Jacobson identities for post-Lie algebras in positive characteristic

Let [Formula: see text] be a prime number. Given a restricted Lie algebra over a field of characteristic [Formula: see text] and a post-Lie operation over it, we prove the Jacobson identities for a [Formula: see text]-structure built from the Lie bracket and the post-Lie operation, called sub-adjacent [Formula: see text]-structure. Furthermore, we give sufficient conditions for the sub-adjacent Lie algebra to be restricted if equipped with this sub-adjacent [Formula: see text]-structure. This construction is “axiomatized” by introducing the notion of restricted post-Lie algebras, and we work out several examples.