  We study classical R-matrices D for Lie algebras L such that D is also a
derivation of L. This yields derivation double Lie algebras (L,D). The
motivation comes from recent work on post-Lie algebra structures on pairs of
Lie algebras arising in the study of nil-affine actions of Lie groups. We prove
that there are no nontrivial simple derivation double Lie algebras, and study
related Lie algebra identities for arbitrary Lie algebras.
