Let $L$ be a central extension of a simple Lie algebra $\mathfrak{g}$ such that $L=[L,L]$. It is not difficult to see that if $H^1(\mathfrak{g}, \mathfrak{g})=0$ then $H^1(L,L)=0$. In other words, if all derivations of $\mathfrak{g}$ are inner, then all derivations of $L$ are inner.

Is the converse true?

For instance, this is indeed the case when $L$ is the universal central extension of $\mathfrak{g}$. (See Theorem 2.2 of [G.M. Benkart - R.V. Moody: Derivations, central extensions, and affine Lie algebras. Algebras Groups Geom. ${\bf 3}$ (1986), no. 4, 456--492.])