It is a special case of the Hurwitz problem, solved also in Radon (1922).
Subsequent proofs of the restrictions on the dimension have been given by Eckmann (1943) using the representation theory of finite groups and by Lee (1948) and Chevalley (1954) using Clifford algebras.
Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups[2] and in quantum mechanics to the classification of simple Jordan algebras.
It is routine to check that the real numbers R, the complex numbers C and the quaternions H are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions.
To show that B ⊕ B j is closed under multiplication Bj = jB.
Imposing the multiplicativity of the norm on C for a + bj and c + dj gives: which leads to Hence d(ac) = (da)c, so that B must be associative.
The proofs of Lee (1948) and Chevalley (1954) use Clifford algebras to show that the dimension N of A must be 1, 2, 4 or 8.
In fact the operators L(a) with (a, 1) = 0 satisfy L(a)2 = −‖a‖2 and so form a real Clifford algebra.
If N is even, N − 1 is odd, so the Clifford algebra has exactly two complex irreducible representations of dimension 2N/2 − 1.
Indeed, taking an orthonormal basis ei of the orthogonal complement of 1 gives rise to operators Ui = L(ei) satisfying This is a projective representation of a direct product of N − 1 groups of order 2.
In fact Eckmann constructed operators of this type in a slightly different but equivalent way.
Let G be the finite group generated by elements vi such that where ε is central of order 2.
The real-valued trace satisfies: These are immediate consequences of the known identities for n = 1.
Since the associators are invariant under cyclic permutations, the diagonal entries of Y are all equal.
To check that Hn(A) satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with (X, X) = Σ ‖xij‖2.
The main axiom to check is the Jordan condition for the operators L(X) defined by L(X)Y = X ∘Y: This is easy to check when A is associative, since Mn(A) is an associative algebra so a Jordan algebra with X ∘Y = 1/2(X Y + Y X).
When A = O and n = 3 a special argument is required, one of the shortest being due to Freudenthal (1951).
With A and n as in the statement of the theorem, let K be the group of automorphisms of E = Hn(A) leaving invariant the inner product.
Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on Mn(A) for any non-associative algebra A.
To prove the diagonalization theorem, take X in E. By compactness k can be chosen in K minimizing the sums of the squares of the norms of the off-diagonal terms of k(X ).
Since the symmetric group Sn, acting by permuting the coordinates, lies in K, if X is not diagonal, it can be supposed that x12 and its adjoint x21 are non-zero.
However, on the line x + y = constant, x2 + y2 has no local maximum (only a global minimum), a contradiction.