On the property of having a finite basis for some varieties of Lie algebras and associative algebras

The following systems of identical relations in Lie and associative algebras are shown to be finitely based. (1) The set of Capelli identities of any given order m. (2) The set of symmetric identities of any given order m. (3) Any set of identities in finitely many variables such that the degrees of all variables but one are bounded. The base field is of characteristic zero.