ABSTRACTIf an algebra satisfies the polynomial identity (for short, is ), then is trivially Lie solvable of index (for short, is ). We prove that the converse holds for subalgebras of the upper triangular matrix algebra any commutative ring, and . We also prove that if a ring S is (respectively, ), then the subring of comprising the upper triangular matrices with constant main diagonal, is (respectively, ) for all . We also study two related questions, namely whether, for a field F, an subalgebra of for some n, with (F-)dimension larger than the maximum dimension of a subalgebra of , exists, and whether a subalgebra of with (the mentioned) maximum dimension, other than the typical subalgebras of with maximum dimension, which were described by Domokos and refined by van Wyk and Ziembowski, exists. Partial results with regard to these two questions are obtained.