I'm trying to understand Weibel's proof that $H^{2}$ classifies extensions of Lie algebras in section 7.6 of Homological algebra. I understand most of the proof until the last section when he shows that the classifying map he has constructed from $\text{Ext}^{1}(g, M) $ to $H^{2}(g, M)$ is injective.
I am not going to write out the full proof, because it is a little bit complicated, but there is one specific part I do not understand.
Weibel defines a $g$-module homomorphism:
$$\tau: f \to e_{1}$$ $$x \mapsto \tau_{1}(x) + D(x) $$
Where $f$ is a free $g$-module, $e_{1}$ is a $g$-module, $ \tau_{1}$ is a $g$-module homomorphism (from $f$ to $e_{1}$), and $D$ is a derivation.
He then argues that $\tau$ is a $g$-module homomorphism as follows:
\begin{align} \tau([x, y]) &= \tau_{1}([x, y]) + D([x, y]) \\ &= [\tau_{1}(x), \tau_{1}(y)] +x(Dy) - y(Dx) \\ &= [\tau_{1}(x) + Dx, \tau_{1}(y) +Dy] \\ &= [\tau(x), \tau(y)] \end{align}
I do not understand why this is true. Why is it the case that:$$ [\tau_{1}(x), \tau_{1}(y)] +x(Dy) - y(Dx) = [\tau_{1}(x) + Dx, \tau_{1}(y) +Dy] ~\text{?}$$
Is this a general fact that holds for any g-module homomorphism $\tau_{1}$ and for any derivation $D$?
Thanks for your help.