Characteristic-p extension of the hypercenter characterization

Determine whether the equality \(Z^*(L)=nil(L)\) from Theorem 3.4(ii) holds for finite-dimensional Lie algebras over fields of prime characteristic.

Background

Theorem 3.4(ii) proves that, over a field of characteristic zero, the nilpotentizer $nil(L)$ equals the hypercenter $Z^*(L)$. The proof uses characteristic-zero tools, including algebraic hulls and Jordan decomposition. The paper does not extend this equality to fields of prime characteristic.

The unresolved issue is therefore whether the characteristic-zero identification remains valid, or requires modification, in positive characteristic.