In the Zorn vector matrix
algebra the three dimensional vector algebra is replaced by a finite dimensional Lie
algebra L over a field of characteristic not 2 equipped with an associative symmetric
bilinear form (a,b) and havin g the property: [a[bc]] = (a,c)b− (a,b)c,a,b,c ∈ L. We
determine all the alternative algebras A obtained in this way: If the bilinear form
(a,b) on L is nondegenerate then A is the split Cayley algebra or a quaternion
algebra. For a degenerate form (a,b), A is a direct sum of its radical and a
subalgebra which is either a quaternion or two dimensional separable algebra. As an
immediate consequence of the first result we have shown that if the bilinear form on
the Lie algebra L is nondegenerate then L is simple with dimension three or
one.
