Let
be an algebra over a field
. For x, y, z in
, write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached
algebra is the same vector space as
, but the product of x and y is x · y. We aim to prove the following result.
THEOREM 1. Let
be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in
, suppose
1![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X0001436X/resource/name/S0008414X0001436X_eqn1.gif?pub-status=live)
Then
is noncommutative Jordan.
The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace
in the Peirce decomposition
. In §4 we construct an ideal of
which we then use to show that the nilpotent elements of
form a subalgebra of
for i = 0, 1.