Skew-symmetric identities in special Lie algebras

In this paper we study the connection between a standard Lie identity and Capelli identities in Lie algebras over a field of characteristic zero. It is shown that the standard identity of degree eight implies all Capelli identities in special Lie algebras, while this is false for degree nine. It is further shown that all Capelli identities follow from a standard identity in a product of two nilpotent varieties provided that the right-hand factor is of class at most two; and that, for $n\geqslant 2$, the Lie algebra $W_n$ satisfies an identity not following from a standard identity.