Subspace property of the nilpotentizer

Determine whether the nilpotentizer \(nil(L)=\{x\in L\mid \langle h,x\rangle\text{ is nilpotent for every }h\in L\}\) is a subspace for every finite-dimensional Lie algebra \(L\).

Background

The nilpotentizer $nil(L)$ is defined as the set of elements that generate a nilpotent subalgebra with every element of the Lie algebra. Earlier results establish inclusions such as $nil(L)\subseteq E(L)$ and identify $nil(L)$ with the hypercenter in characteristic zero, but they do not establish that $nil(L)$ is closed under addition and scalar multiplication in arbitrary characteristic.

The question asks for a general structural characterization of this set, specifically whether it always has the linear-subspace structure naturally suggested by its definition.