Some Lie algebra structures on symmetric powers

Let k be a field of any characteristic, V a finite-dimensional vector space over k, and Sᵈ(V^*) be the d-th symmetric power of the dual space V^*. Given a linear map φ on V and an eigenvector w of φ, we prove that the pair (φ, w) can be used to construct a new Lie algebra structure on Sᵈ(V^*). We prove that this Lie algebra structure is solvable, and in particular, it is nilpotent if φ is a nilpotent map. We also classify the Lie algebras for all possible pairs (φ, w), when k=C and V is two-dimensional.