On special Lie algebras having a faithful module with Krull dimension

For special Lie algebras we prove an analogue of Markov's theorem on $\mathrm{PI}$-algebras having a faithful module with Krull dimension: the solubility of the prime radical. We give an example of a semiprime Lie algebra that has a faithful module with Krull dimension but cannot be represented as a subdirect product of finitely many prime Lie algebras. We prove a criterion for a semiprime Lie algebra to be representable as such a subdirect product.