The commutator
未d−d未 of two derivations
未,d always yields a derivation and must be inner if one of
未,d is inner. We generalize this to primitive expressions of skew derivations (
Theorem 3.2 and
Theorem 3.3). Our main results are
Theorem 2.6 and
Theorem 2.7. As applications, we first apply these to quantum Lie operations (
Theorem 3.6). Then we extend Koryukin's counterexample
[14] by adding primitive identities involving inner skew derivations (
Theorem 3.7). We work in the setting of Hopf algebras and symmetric smash products.
Theorem 1.3 is very useful in analyzing identities in this context.