For the class \(T(M)\) of nilpotent Lie algebras which are isomorphic to upper triangular \(M\times M\) matrices, the invariants are explicitly given (by working in the coadjoint representation). There are \([\frac M2]\) independent polynomial invariants.
In an earlier paper [J. Phys. A, Math. Gen. 31, 789-806 (1998; Zbl 1001.17011)] the authors studied the class \(L(M,f)\) of solvable Lie algebras which have \(T(M)\) as their nilradical and \(f\) \((1\leq f\leq M-1)\) additional linearly nilindependent elements. Here, all invariants for \(L(4,f)\) are given. They cannot all be chosen to be polynomials. For arbitrary \(M\) and \(f=1\) and \(f=M-1\), invariants are given (without a proof of their completeness).