On similarity invariants of matrix commutators

We study the possible eigenvalues, ranks and numbers of nonconstant invariant polynomials of [ [[A,X1],X2],…,Xk], when A is a fixed matrix and X1,…,Xk vary. Then we generalize these results in the following way. Let g(X1,…, Xk) be any expression obtained from distinct noncommuting variables X1,…,Xk by applying recursively the Lie product [• ,•] and without using the same variable twice. We study the possible eigenvalues, ranks and numbers of nonconstant invariant polynomials of g(X1,…,Xk) when one of the variables X1,…,Xk takes a fixed value in Fn×n and the others vary.