A $\delta$-first Whitehead Lemma

We prove that $\delta$-derivations of a simple finite-dimensional Lie algebra over a field of characteristic zero, with values in a finite-dimensional module, are either inner derivations, or, in the case of adjoint module, multiplications by a scalar, or some exceptional cases related to $\mathfrak{s}\mathfrak{l}(2)$. This can be viewed as an extension of the classical first Whitehead Lemma.