The ``measure of non-Lieness'' for Mal'tsev algebras (Q1317608)

Let \(\Phi\) be a commutative-associative ring with 1/6, and \(A\) be a Mal'tsev \(\Phi\)-algebra. If \(A\) can be isomorphically embedded in the commutator algebra \(B^{(-)}\) of some alternative algebra \(B\), then \(A\) is said to be special, and \(B\) is called an alternative enveloping algebra for \(A\). Let \(J(A)\) denote the ideal generated in \(A\) by all Jacobians \(J(x,y,z)= (xy)z+ (yz)x+ (zx)y\), where \(x,y,z\in A\). The author proves that \(J(A)\) is special; and if \(\Phi\) is a field such that \(J(A)\) is finite-dimensional over \(\Phi\), then an alternative enveloping algebra for \(J(A)\) is also finite-dimensional. The proof of this result involves the construction of an invariant symmetric bilinear form on \(J(A)\) over a commutative subalgebra \(K\) of the centroid of \(J(A)\). It is also proved that \(J(A)\) is Shirshov locally-finite over \(K\).