Some Lie algebra structures on symmetric powers

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