View source for Hurwitz's theorem (composition algebras)

{{Short description|Non-associative algebras with positive-definite quadratic form}}
In [[mathematics]], '''Hurwitz's theorem''' is a [[theorem]] of [[Adolf Hurwitz]] (1859–1919), published posthumously in 1923, solving the [[Hurwitz problem]] for [[dimension (vector space)|finite-dimensional]] [[unital algebra|unital]] [[real number|real]] [[non-associative algebra]]s endowed with a nondegenerate [[definite quadratic form|positive-definite]] [[quadratic form]]. The theorem states that if the quadratic form defines a [[homomorphism]] into the positive real numbers on the non-zero part of the algebra, then the algebra must be [[isomorphic]] to the [[real number]]s, the [[complex number]]s, the [[quaternion]]s, or the [[octonion]]s, and that there are no other possibilities. Such algebras, sometimes called '''Hurwitz algebras''', are examples of [[composition algebra]]s.

The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary [[field (mathematics)|field]]s.<ref>See:
*{{harvnb|Lam|2005}}
*{{harvnb|Rajwade|1993}}
*{{harvnb|Shapiro|2000}}</ref> Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally [[mathematical proof|proved]] by Hurwitz in 1898. It is a special case of the [[Hurwitz problem]], solved also in {{harvtxt|Radon|1922}}. Subsequent proofs of the restrictions on the dimension have been given by {{harvtxt|Eckmann|1943}} using the [[representation theory of finite groups]] and by  {{harvtxt|Lee|1948}} and {{harvtxt|Chevalley|1954}} using [[Clifford algebra]]s. Hurwitz's theorem has been applied in [[algebraic topology]] to problems on [[vector fields on spheres#Radon–Hurwitz numbers|vector fields on spheres]] and the [[homotopy group]]s of the [[classical group]]s<ref>See:
*{{harvnb|Eckmann|1989}}
*{{harvnb|Eckmann|1999}}</ref> and in [[quantum mechanics]] to the [[Symmetric cone#Simple Euclidean Jordan algebras|classification of simple Jordan algebras]].<ref>{{harvnb|Jordan|von Neumann|Wigner|1934}}</ref>

==Euclidean Hurwitz algebras==

===Definition===
A '''Hurwitz algebra''' or '''composition algebra''' is a finite-dimensional not necessarily associative algebra {{mvar|A}} with identity endowed with a nondegenerate quadratic form {{mvar|q}} such that {{math|1=''q''(''a b'') = ''q''(''a'') ''q''(''b'')}}. If the underlying coefficient field is the reals and {{mvar|q}} is positive-definite, so that {{math|1=(''a'', ''b'') = {{sfrac|1|2}}[''q''(''a'' + ''b'') − ''q''(''a'') − ''q''(''b'')]}} is an [[inner product space|inner product]], then {{mvar|A}} is called a '''Euclidean Hurwitz algebra''' or (finite-dimensional) '''normed division algebra'''.<ref>{{harvnb|Faraut|Koranyi|1994|p=82}}</ref>

If {{mvar|A}} is a Euclidean Hurwitz algebra and {{mvar|a}} is in {{mvar|A}}, define the [[involution (mathematics)|involution]] and right and left multiplication operators by

:<math>a^* = -a + 2(a,1)1,\quad L(a)b = ab,\quad R(a)b = ba.</math>

Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties:

* the involution is an [[antiautomorphism]], i.e. {{math|1=(''ab'')* = ''b''*''a''*}}
* {{math|1=''aa''* = ‖''a''‖<sup>2</sup> 1 = ''a''*''a''}}
* {{math|1=''L''(''a''*) = ''L''(''a'')*}}, {{math|1=''R''(''a''*) = ''R''(''a'')*}}, so that the involution on the algebra corresponds to taking [[adjoint operator|adjoints]]
* {{math|1=Re (''ab'') = Re (''ba'')}} if {{math|1=Re ''x'' = (''x'' + ''x''*)/2 = (''x'', 1)1}}
* {{math|1=Re (''ab'')''c'' = Re ''a''(''bc'')}}
* {{math|1=''L''(''a''<sup>2</sup>) = ''L''(''a'')<sup>2</sup>}}, {{math|1=''R''(''a''<sup>2</sup>) = ''R''(''a'')<sup>2</sup>}}, so that {{mvar|A}} is an [[alternative algebra]].

These properties are proved starting from the polarized version of the identity {{math|1=(''ab'', ''ab'') = (''a'', ''a'')(''b'', ''b'')}}:

:<math>\displaystyle{2(a,b)(c,d) = (ac,bd) + (ad,bc).}</math>

Setting {{math|1=''b'' = 1}} or {{math|1=''d'' = 1}} yields {{math|1=''L''(''a''*) = ''L''(''a'')*}} and {{math|1=''R''(''c''*) = ''R''(''c'')*}}.

Hence {{math|1=Re(''ab'') = (''ab'', 1)1 = (''a'', ''b''*)1 = (''ba'', 1)1 = Re(''ba'')}}.

Similarly {{math|1= Re (''ab'')''c'' = ((''ab'')''c'',1)1 = (''ab'', ''c''*)1 = (''b'', ''a''* ''c''*)1 = (''bc'',''a''*)1 = (''a''(''bc''),1)1 = Re ''a''(''bc'')}}.

Hence {{math|1= ((''ab'')*, c) = (''ab'', ''c''*) = (''b'', ''a''*''c''*) = (1, ''b''*(''a''*''c''*)) = (1, (''b''*''a''*)''c''*) = (''b''*''a''*, ''c'')}}, so that {{math|1=(''ab'')* = ''b''*''a''*}}.

By the polarized identity {{math|1=‖''a''‖<sup>2</sup> (''c'', ''d'') = (''ac'', ''ad'') = (''a''* (''ac''), ''d'')}} so {{math|1=''L''(''a''*) ''L''(''a'') = ''L''(‖''a''‖<sup>2</sup>)}}. Applied to 1 this gives {{math|1=''a''*''a'' = ‖''a''‖<sup>2</sup> 1}}. Replacing {{mvar|a}} by {{math|''a''*}} gives the other identity.

Substituting the formula for {{math|''a''*}} in {{math|1=''L''(''a''*) ''L''(''a'') = ''L''(''a''*''a'')}} gives {{math|1=''L''(''a'')<sup>2</sup> = ''L''(''a''<sup>2</sup>)}}. The formula {{math|1=''R''(''a''<sup>2</sup>) = ''R''(''a'')<sup>2</sup>}} is proved analogously.

===Classification===
It is routine to check that the real numbers {{math|'''R'''}}, the complex numbers {{math|'''C'''}} and the quaternions {{math|'''H'''}} are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions {{math|'''R''' ⊂ '''C''' ⊂ '''H'''}}.

Analysing such an inclusion leads to the [[Cayley–Dickson construction]], formalized by [[Abraham Adrian Albert|A.A. Albert]]. Let {{mvar|A}} be a Euclidean Hurwitz algebra and {{mvar|B}} a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a [[unit vector]] {{mvar|j}} in {{mvar|A}} orthogonal to {{mvar|B}}. Since {{math|1=(''j'', 1) = 0}}, it follows that {{math|1=''j''* = −''j''}} and hence {{math|1=''j''<sup>2</sup> = −1}}. Let {{mvar|C}} be subalgebra generated by {{mvar|B}} and {{mvar|j}}. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following [[Cayley–Dickson construction|'''Cayley–Dickson multiplication laws''']]:

:<math>\displaystyle{C=B\oplus Bj, \,\,\, (a+bj)^*=a^* - bj, \,\,\, (a+bj)(c+dj)=(ac -d^*b) +(bc^*+da)j.}</math>

{{mvar|B}} and {{math|''Bj''}} are orthogonal, since {{mvar|j}} is orthogonal to {{mvar|B}}. If {{mvar|a}} is in {{mvar|B}}, then {{math|1=''j a'' = ''a''* ''j''}}, since by orthogonal {{math|1=0 = 2(''j'', ''a''*) = ''ja'' − ''a''*''j''}}. The formula for the involution follows. To show that {{math|''B'' ⊕ ''B j''}} is closed under multiplication {{math|1=''Bj'' = ''jB''}}. Since {{math|''Bj''}} is orthogonal to 1, {{math|1=(''bj'')* = −''bj''}}.

* {{math|1=''b''(''cj'') = (''cb'')&hairsp;''j''}} since {{math|1=(''b'', ''j'') = 0}} so that, for {{mvar|x}} in {{mvar|A}}, {{math|1=(''b''(''cj''), ''x'') = (''b''(&hairsp;''jx''), ''j''(''cj'')) = −(''b''(&hairsp;''jx''), ''c''*) = −(''cb'', (&hairsp;''jx'')*) = −((''cb'')&hairsp;''j'', ''x''*) = ((''cb'')&hairsp;''j'', ''x'')}}.
* {{math|1=(&hairsp;''jc'')''b'' = ''j''(''bc'')}} taking adjoints above.
* {{math|1=(''bj'')(''cj'') = −''c''*''b''}} since {{math|1=(''b'', ''cj'')}} = 0, so that, for {{mvar|x}} in {{mvar|A}}, {{math|1=((''bj'')(''cj''), ''x'') = −((''cj'')''x''*, ''bj'') = (''bx''*, (''cj'')&hairsp;''j'') = −(''c''*''b'', ''x'')}}.

Imposing the multiplicativity of the norm on {{mvar|C}} for {{math|''a'' + ''bj''}} and {{math|''c'' + ''dj''}} gives:

:<math>\displaystyle{(\|a\|^2 + \|b\|^2)(\|c\|^2 + \|d\|^2) = \|ac - d^*b\|^2 + \|bc^* + da\|^2,}</math>

which leads to

:<math>\displaystyle{(ac,d^*b) = (bc^*,da).}</math>

Hence {{math|1=''d''(''ac'') = (''da'')''c''}}, so that {{mvar|B}} ''must be associative''.

This analysis applies to the inclusion of {{math|1='''R'''}} in {{math|1='''C'''}} and {{math|1='''C'''}} in {{math|1='''H'''}}. Taking {{math|1='''O''' = '''H''' ⊕ '''H'''}} with the product and inner product above gives a noncommutative nonassociative algebra generated by {{math|1='''J''' = (0, 1)}}. This recovers the usual definition of the [[octonion]]s or [[Cayley number]]s. If {{mvar|A}} is a Euclidean algebra, it must contain {{math|'''R'''}}. If it is strictly larger than {{math|'''R'''}}, the argument above shows that it contains {{math|'''C'''}}. If it is larger than {{math|'''C'''}}, it contains {{math|'''H'''}}. If it is larger still, it must contain {{math|'''O'''}}. But there the process must stop, because {{math|'''O'''}} is not associative. In fact {{math|'''H'''}} is not commutative and {{math|1=''a''(''bj'') = (''ba'')&hairsp;''j'' ≠ (''ab'')&hairsp;''j''}} in {{math|1='''O'''}}.<ref>{{harvnb|Faraut|Koranyi|1994|pp=81–86}}</ref>

'''{{smallcaps|Theorem.}}''' The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.

==Other proofs==
The proofs of {{harvtxt|Lee|1948}} and {{harvtxt|Chevalley|1954}} use [[Clifford algebra]]s to show that the dimension {{mvar|N}} of {{mvar|A}} must be 1, 2, 4 or 8. In fact the operators {{math|''L''(''a'')}} with {{math|1=(''a'', 1) = 0}} satisfy {{math|1=''L''(''a'')<sup>2</sup> = −‖''a''‖<sup>2</sup>}} and so form a real Clifford algebra. If {{mvar|a}} is a unit vector, then {{math|''L''(''a'')}} is skew-adjoint with square {{math|−''I''}}. So {{mvar|N}} must be either [[parity (mathematics)|even]] or 1 (in which case {{mvar|A}} contains no unit vectors orthogonal to 1).  The real Clifford algebra and its [[complexification]] act on the complexification of {{mvar|A}}, an {{mvar|N}}-dimensional complex space.  If {{mvar|N}} is even, {{math|''N'' − 1}} is [[parity (mathematics)|odd]], so the Clifford algebra has exactly two complex [[irreducible representation]]s of dimension {{math|2<sup>''N''/2 − 1</sup>}}. So this [[power of 2]] must divide {{mvar|N}}. It is easy to see that this implies {{mvar|N}} can only be 1, 2, 4 or 8.

The proof of {{harvtxt|Eckmann|1943}} uses the [[representation theory]] of [[finite group]]s, or the projective representation theory of [[elementary abelian 2-group]]s, known to be equivalent to the representation theory of real Clifford algebras. Indeed, taking an [[orthonormal basis]] {{math|''e''<sub>''i''</sub>}} of the [[orthogonal complement]] of 1 gives rise to operators {{math|1=''U''<sub>''i''</sub> = ''L''(''e''<sub>''i''</sub>)}}
satisfying

:<math>U_i^2 = -I,\quad U_i U_j = -U_j U_i \,\, (i \ne j).</math>

This is a [[projective representation]] of a direct product of {{math|''N'' − 1}} [[group (mathematics)|groups]] of [[order of a group|order]] 2. ({{mvar|N}} is assumed to be greater than 1.) The operators {{math|''U''<sub>''i''</sub>}} by construction are skew-symmetric and orthogonal. In fact Eckmann constructed operators of this type in a slightly different but equivalent way. It is in fact the method originally followed in {{harvtxt|Hurwitz|1923}}.<ref>See:
*{{harvnb|Hurwitz|1923|p=11}}
*{{harvnb|Herstein|1968|pp=141–144}}</ref> Assume that there is a composition law for two forms

:<math>\displaystyle{(x_1^2 + \cdots +x_N^2)(y_1^2 + \cdots + y_N^2) =z_1^2 + \cdots + z_N^2,}</math>

where {{math|''z''<sub>''i''</sub>}} is bilinear in {{mvar|x}} and {{mvar|y}}. Thus

:<math>\displaystyle{z_i=\sum_{j=1}^N a_{ij}(x)y_j}</math>

where the [[matrix (mathematics)|matrix]] {{math|1=''T''(''x'') = (''a''<sub>''ij''</sub>)}} is linear in {{mvar|x}}. The relations above are equivalent to

:<math>\displaystyle{T(x)T(x)^t=x_1^2 +\cdots + x_N^2.}</math>

Writing

:<math>\displaystyle{T(x)=T_1x_1 + \cdots + T_Nx_N,}</math>

the relations become

:<math>\displaystyle{T_iT^t_j+T_jT_i^t =2\delta_{ij}I.}</math>

Now set {{math|1=''V''<sub>''i''</sub> = (''T''<sub>''N''</sub>)<sup>''t''</sup> ''T''<sub>''i''</sub>}}. Thus {{math|1=''V''<sub>''N''</sub> = ''I''}} and the {{math|''V''<sub>1</sub>, ... , ''V''<sub>''N'' − 1</sub>}} are skew-adjoint, orthogonal satisfying exactly the same relations as the {{math|''U''<sub>''i''</sub>}}'s:

:<math>\displaystyle{V_i^2 = -I,\quad V_iV_j = -V_jV_i \,\, (i \ne j).}</math>

Since {{math|''V''<sub>''i''</sub>}} is an [[orthogonal matrix]] with square {{math|−''I''}} on a real [[vector space]], {{mvar|N}} is even.

Let {{mvar|G}} be the finite group generated by elements {{math|''v''<sub>''i''</sub>}} such that

:<math>\displaystyle{v_i^2 = \varepsilon,\quad v_iv_j = \varepsilon v_jv_i \,\, (i \ne j),}</math>

where {{mvar|ε}} is [[center (group theory)|central]] of order 2. The [[commutator subgroup]] {{math|[''G'', ''G'']}} is just formed of 1 and {{mvar|ε}}. If {{mvar|N}} is odd this coincides with the [[center (group theory)|center]] while if {{mvar|N}} is even the center has order 4 with extra elements {{math|1=γ = ''v''<sub>1</sub>...''v''<sub>''N'' −&thinsp;1</sub>}} and {{math|''ε''γ}}. If {{mvar|g}} in {{mvar|G}} is not in the center its [[conjugacy class]] is exactly {{mvar|g}} and {{math|''εg''}}. Thus there are
{{math|2<sup>''N'' −&thinsp;1</sup> + 1}} conjugacy classes for {{mvar|N}} odd and {{math|2<sup>''N'' −&thinsp;1</sup> + 2}} for {{mvar|N}} even. {{mvar|G}} has {{math|1={{!}}&hairsp;{{mvar|G}} / [''G'', ''G'']&thinsp;{{!}} = 2<sup>''N'' −&thinsp;1</sup>}} 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since {{mvar|N}} is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals {{math|{{!}}''G''{{!}}}} and the dimensions divide {{math|{{!}}''G''{{!}}}}, the two irreducibles must have dimension {{math|2<sup>(''N'' − 2)/2</sup>}}. When {{mvar|N}} is even, there are two and their dimension must divide the order of the group, so is a power of two, so they must both have dimension {{math|2<sup>(''N'' − 2)/2</sup>}}. The space on which the {{math|''V''<sub>''i''</sub>}}'s act can be complexified. It will have complex dimension {{mvar|N}}. It breaks up into some of complex irreducible representations of {{mvar|G}}, all having dimension {{math|2<sup>(''N'' − 2)/2</sup>}}. In particular this dimension is {{math|≤ ''N''}}, so {{mvar|N}} is less than or equal to 8. If {{math|1=''N'' = 6}}, the dimension is 4, which does not divide 6. So ''N'' can only be 1, 2, 4 or 8.

==Applications to Jordan algebras==
Let {{mvar|A}} be a Euclidean Hurwitz algebra and let {{math|''M''<sub>''n''</sub>(''A'')}} be the algebra of {{mvar|n}}-by-{{mvar|n}} matrices over {{mvar|A}}. It is a unital nonassociative algebra with an involution given by

:<math>\displaystyle{(x_{ij})^*=(x_{ji}^*).}</math>

The trace {{math|Tr(''X''&hairsp;)}} is defined as the sum of the diagonal elements of {{mvar|X}} and the real-valued trace by
{{math|1=Tr<sub>'''R'''</sub>(''X''&hairsp;) = Re Tr(''X''&hairsp;)}}. The real-valued trace satisfies:

:<math>\operatorname{Tr}_{\mathbf{R}} XY = \operatorname{Tr}_{\mathbf{R}} YX, \qquad \operatorname{Tr}_{\mathbf{R}} (XY)Z = \operatorname{Tr}_{\mathbf{R}} X(YZ).</math>

These are immediate consequences of the known identities for {{math|1=''n'' = 1}}.

In {{mvar|A}} define the ''[[associator]]'' by

:<math>\displaystyle{[a,b,c]=a(bc) - (ab)c.}</math>

It is trilinear and vanishes identically if {{mvar|A}} is associative. Since {{mvar|A}} is an [[alternative algebra]]
{{math|1=[''a'', ''a'', ''b''] = 0}} and {{math|1=[''b'', ''a'', ''a''] = 0}}. Polarizing it follows that the associator is antisymmetric in its three entries.  Furthermore, if {{mvar|a}}, {{mvar|b}} or {{mvar|c}} lie in {{math|'''R'''}} then {{math|1=[''a'', ''b'', ''c''] = 0}}. These facts imply that {{math|''M''<sub>3</sub>(''A'')}} has certain commutation properties. In fact if {{mvar|X}} is a matrix in {{math|''M''<sub>3</sub>(''A'')}} with real entries on the diagonal then

:<math>\displaystyle{[X,X^2] = aI,}</math>

with {{mvar|a}} in {{mvar|A}}. In fact if {{math|1=''Y'' = [''X'',&thinsp; ''X''<sup>&thinsp;2</sup>]}}, then

:<math>\displaystyle{y_{ij} = \sum_{k,\ell} [x_{ik},x_{k\ell},x_{\ell j}].}</math>

Since the diagonal entries of {{mvar|X}} are real, the off-diagonal entries of {{mvar|Y}} vanish. Each diagonal
entry of {{mvar|Y}} is a sum of two associators involving only off diagonal terms of {{mvar|X}}. Since the associators are invariant under [[cyclic permutation]]s, the diagonal entries of {{mvar|Y}} are all equal.

Let {{math|''H''<sub>''n''</sub>(''A'')}} be the space of self-adjoint elements in {{math|''M''<sub>''n''</sub>(''A'')}} with product {{math|1=''X''&hairsp;∘''Y'' = {{sfrac|1|2}}(''X Y'' + ''Y X'')}} and inner product {{math|1=(''X'', ''Y''&hairsp;) = Tr<sub>'''R'''</sub>(''X Y''&hairsp;)}}.

'''{{smallcaps|Theorem.}}''' {{math|''H''<sub>''n''</sub>(''A'')}} is a [[Euclidean Jordan algebra]] if {{mvar|A}} is associative (the real numbers, complex numbers or quaternions) and {{math|1=''n'' ≥ 3}} or if {{mvar|A}} is nonassociative (the octonions) and {{math|1=''n'' = 3}}.

The [[exceptional object|exceptional]] Jordan algebra {{math|''H''<sub>3</sub>('''O''')}} is called the '''[[Albert algebra]]''' after [[Abraham Adrian Albert|A.A. Albert]].

To check that {{math|''H''<sub>''n''</sub>(''A'')}} satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a [[symmetric bilinear form]] with {{math|1=(''X'', ''X'') = Σ&thinsp;‖''x''<sub>''ij''</sub>‖<sup>2</sup>}}. So it is an inner product. It satisfies the associativity property {{math|1=(''Z''∘''X'', ''Y''&hairsp;) = (''X'', ''Z''∘''Y''&hairsp;)}} because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators {{math|''L''(''X'')}} defined by {{math|1=''L''(''X'')''Y'' = ''X''&hairsp;∘''Y''}}:

:<math>\displaystyle{[L(X),L(X^2)]=0.}</math>

This is easy to check when {{mvar|A}} is associative, since {{math|''M''<sub>''n''</sub>(''A'')}} is an associative algebra so a Jordan algebra with {{math|1=''X''&hairsp;∘''Y'' = {{sfrac|1|2}}(''X Y'' + ''Y X'')}}. When {{math|1=''A'' = '''O'''}} and {{math|1=''n'' = 3}} a special argument is required, one of the shortest being due to {{harvtxt|Freudenthal|1951}}.<ref>
See:
*{{harvnb|Faraut|Koranyi|1994|pp=88–91}}
*{{harvnb|Postnikov|1986}}</ref>

In fact if {{mvar|T}} is in {{math|''H''<sub>3</sub>('''O''')}} with {{math|1=Tr ''T'' = 0}}, then

:<math>\displaystyle{D(X) = TX -XT}</math>

defines a skew-adjoint derivation of {{math|''H''<sub>3</sub>('''O''')}}. Indeed,

:<math>\operatorname{Tr}(T(X(X^2)) -T(X^2(X))) = \operatorname{Tr} T(aI) = \operatorname{Tr}(T)a=0,</math>

so that

:<math>(D(X),X^2) = 0.</math>

Polarizing yields:

:<math>(D(X),Y\circ Z) + (D(Y),Z\circ X) + (D(Z),X\circ Y) = 0.</math>

Setting {{math|1=''Z'' = 1}} shows that {{mvar|D}} is skew-adjoint. The derivation property {{math|1=''D''(''X''&thinsp;∘''Y'') = ''D''(''X'')∘''Y'' + ''X''∘''D''(''Y'')}} follows by this and the associativity property of the inner product in the identity above.

With {{mvar|A}} and {{mvar|n}} as in the statement of the theorem, let {{mvar|K}} be the group of [[automorphism]]s of {{math|1=''E'' = ''H''<sub>''n''</sub>(''A'')}} leaving invariant the inner product. It is a closed subgroup of {{math|1=[[orthogonal group|O]](''E'')}} so a [[compact space|compact]] [[Lie group]]. Its [[Lie algebra]] consists of skew-adjoint derivations. {{harvtxt|Freudenthal|1951}} showed that given {{mvar|X}} in {{mvar|E}} there is an automorphism {{mvar|k}} in {{mvar|K}} such that {{math|''k''(''X'')}} is a [[diagonal matrix]]. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on {{math|''M''<sub>''n''</sub>(''A'')}} for any non-associative algebra {{mvar|A}}.

To prove the diagonalization theorem, take {{mvar|X}} in {{mvar|E}}. By compactness {{mvar|k}} can be chosen in {{mvar|K}} minimizing the sums of the squares of the norms of the off-diagonal terms of {{math|''k''(''X''&hairsp;)}}. Since {{mvar|K}} preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms of {{math|''k''(''X''&hairsp;)}}. Replacing {{mvar|X}} by {{math|''k X''}}, it can be assumed that the maximum is attained at {{mvar|X}}. Since the [[symmetric group]] {{math|''S''<sub>''n''</sub>}}, acting by permuting the coordinates, lies in {{mvar|K}}, if {{mvar|X}} is not diagonal, it can be supposed that {{math|''x''<sub>12</sub>}} and its adjoint {{math|''x''<sub>21</sub>}} are non-zero. Let {{mvar|T}} be the skew-adjoint matrix with {{math|(2, 1)}} entry {{mvar|a}}, {{math|(1, 2)}} entry {{math|−''a''*}} and 0 elsewhere and let {{mvar|D}} be the derivation ad {{mvar|T}} of {{mvar|E}}. Let {{math|1=''k''<sub>''t''</sub> = exp ''tD''}} in {{mvar|K}}. Then only the first two diagonal entries in {{math|1=''X''(''t'') = ''k''<sub>''t''</sub>''X''}} differ from those of {{mvar|X}}. The diagonal entries are real. The derivative of {{math|''x''<sub>11</sub>(''t'')}} at {{math|1=''t'' = 0}} is the {{math|(1, 1)}} coordinate of {{math|[''T'', ''X'']}}, i.e. {{math|1=''a''* ''x''<sub>21</sub> + ''x''<sub>12</sub> ''a'' = 2(''x''<sub>21</sub>, ''a'')}}. This derivative is non-zero if {{math|1=''a'' = ''x''<sub>21</sub>}}. On the other hand, the group {{math|''k''<sub>''t''</sub>}} preserves the real-valued trace. Since it can only change {{math|1=''x''<sub>11</sub>}} and {{math|1=''x''<sub>22</sub>}}, it preserves their sum. However, on the line {{math|1=''x'' + ''y'' =}} constant, {{math|''x''<sup>2</sup> + ''y''<sup>2</sup>}} has no local maximum (only a global minimum), a contradiction. Hence {{mvar|X}} must be diagonal.

==See also==

* [[Multiplicative quadratic form]]
* [[Radon–Hurwitz number]]
* [[Frobenius theorem (real division algebras)|Frobenius Theorem]]

==Notes==
{{reflist|24em}}

==References==
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==Further reading==
*{{citation|last=Baez|first= John C.|title= The octonions|journal= Bull. Amer. Math. Soc.|volume= 39|issue= 2| year=2002|pages= 145–205|url=http://math.ucr.edu/home/baez/octonions/|doi=10.1090/S0273-0979-01-00934-X|arxiv=math/0105155|s2cid= 586512}}
* {{citation|last1=Conway| first1=John H.|last2= Smith|first2= Derek A.|title= On quaternions and octonions: their geometry, arithmetic, and symmetry|publisher=A K Peters|year= 2003|isbn= 978-1568811345}}
*{{citation |title=Hypercomplex numbers. An elementary introduction to algebras |year=1989 |publisher=[[Springer-Verlag]] |first1=I.L. |last1=Kantor |first2=A.S. |last2=Solodovnikov |others=Trans. A. Shenitzer |page=[https://archive.org/details/hypercomplexnumb0000kant/page/121 121] |chapter=Normed algebras with an identity. Hurwitz's theorem. |edition=2nd |isbn=978-0-387-96980-0 |zbl=0669.17001 |chapter-url=https://archive.org/details/hypercomplexnumb0000kant/page/121 }}
* [[Max Koecher]] & [[Reinhold Remmert]] (1990) "Composition Algebras. Hurwitz's Theorem — Vector-Product Algebras", chapter 10 of ''Numbers'' by [[Heinz-Dieter Ebbinghaus]] et al., Springer, {{ISBN|0-387-97202-1}}
*{{citation |first=T. A. |last=Springer |author-link=T. A. Springer |author2=F. D. Veldkamp |year=2000 |title=Octonions, Jordan Algebras and Exceptional Groups |publisher=[[Springer-Verlag]] | isbn= 978-3-540-66337-9}}

[[Category:Composition algebras]]
[[Category:Non-associative algebras]]
[[Category:Quadratic forms]]
[[Category:Representation theory]]
[[Category:Theorems about algebras]]
[[Category:1923 introductions]]
[[Category:Hypercomplex numbers]]
[[Category:Octonions]]