ALGEBRAS SATISFYING THE POLYNOMIAL IDENTITY [x1, x2][x3, x4, x5]=0

Let K be a field of characteristic zero, and M-5 the variety of associative unitary algebras defined by the polynomial identity [x(1), x(2)][x(3), x(4), x(5)] = 0. This variety is one of the several minimal varieties of exponent 3 (and all proper subvarieties are of exponents 1 and 2). We describe asymptotically its proper subvarieties. More precisely, we define certain algebras R-2k for any k is an element of N and show that if U is a proper subvariety of M-5, then the T-ideal of its polynomial identities is asymptotically equivalent to the T-ideal of the identities of one of the algebras K, E, R-2k or R-2k circle plus E, for a suitable k is an element of N. We give also another description relating the T-ideals of the proper subvarieties of M-5 with the polynomial identities of upper triangular matrices of a suitable size.