Lie solvability in matrix algebras

If an algebra A satisfies the polynomial identity x1,y1x2,y2⋯x2m,y2m=0(for short, A is D2m), then A is trivially Lie solvable of index m+1 (for short, A is Lsm+1). We prove that the converse holds for subalgebras of the upper triangular matrix algebra Un(R),R any commutative ring, and n≥1. We also prove that if a ring S is D2 (respectively, Ls2), then the subring Um⋆(S) of Um(S) comprising the upper triangular m×m matrices with constant main diagonal, is D2log2m (respectively, Lslog2m+1) for all m≥2. We also study two related questions, namely whether, for a field F, an Ls2 subalgebra of Mn(F), for some n, with (F-)dimension larger than the maximum dimension 2+3n28 of a D2 subalgebra of Mn(F), exists, and whether a D2 subalgebra of Un(F) with (the mentioned) maximum dimension, other than the typical D2 subalgebras of Un(F) 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.