Non-linear maps preserving solvability

Let Mn be the algebra of all n×n complex matrices and let L be the general linear Lie algebra gl(n,C) or the special linear Lie algebra sl(n,C). A bijective (not necessarily linear) map preserves solvability in both directions if both and −1 map every solvable Lie subalgebra of L into some solvable Lie subalgebra. If n 3 then every such map is either a composition of a bijective lattice preserving map with a similarity transformation and a map [aij] [f(aij)] induced by a field automorphism , or a map of this type composed with the transposition. We also describe the general form of such maps in the case when n=2. Using Lieʹs theorem we will reduce the proof of this statement to the problem of characterizing bijective maps on Mn preserving triangularizability of matrix pairs in both directions. As a byproduct we will characterize bijective maps on Mn that preserve inclusion for lattices of invariant subspaces in both directions.