On the maximum dimensions of subalgebras of M (K) satisfying two related identities

For an arbitrary q≥2, we find an upper bound for the dimension of a subalgebra of the full matrix algebra Mn(K) over an arbitrary field K satisfying the identity[x1,y1,z1]⋅[x2,y2,z2]⋅⋯⋅[xq,yq,zq]=0, and we show that this upper bound is sharp by presenting an example in block triangular form of a subalgebra of Mn(K) with dimension equal to the obtained upper bound. We apply this result to Lie solvable algebras of index 2, i.e., algebras satisfying the identity [x1,y1,x2,y2]=0. To be precise, for n≤4, we find the sharp upper bound for the dimension of a Lie solvable subalgebra of Mn(K) of index 2, and for n>4, we obtain the relatively tight (at least for small values of n>4) interval2+⌊3n28⌋,2+⌊5n212⌋ for the maximum dimension of a Lie solvable subalgebra of Mn(K) of index 2, the exact value of which is not known.