On semialternative algebras
A nonassociative ring satisfying the identity \((x,y,z)= (y,z,x)\) is called semialternative. The authors assume finite dimensionality and characteristic \(\neq 2,3\). They define the radical as the maximal solvable ideal and show that this radical is nilpotent. If the ring has an idempotent, they find its Peirce decomposition and subspace multiplication table. They have an example which shows that the Wedderburn principal theorem does not hold. Over a field of characteristic 0, they define a trace form \((x,y):=\) the trace of right multiplication by \(xy\). Then \((xy,z)=(x,yz)\) and the radical of the bilinear form is the same as the radical previously defined.
scientific article; zbMATH DE number 130497