The notion of strict Lie 2-algebra is the special case of a general notion Lie 2-algebra for which the Jacobi identity does hold (and not just up to nontrivial isomorphism).

More precisely: a strict Lie 2-algebra is an ∞-Lie algebra with generators just in degree 1 and 2 and at most the unary and binary brackets being nontrivial.

Equivalently, this is a dg-Lie algebra with generators in the lowest two degrees.

The Lie version of a smooth automorphism 2-group is the derivation Lie 2-algebra $Der(\mathfrak{g})$ of an ordinary Lie algebra$\mathfrak{g}$. This is the one coming from the differential crossed module$(\mathfrak{g} \stackrel{ad}{\to} der(\mathfrak{g}))$.