 First-order algebraic differential operator decomposition

If $\theta$ has order at most one, set $c=\theta (1)$ and $\delta =\theta -m_c$. Then $\delta (1)=0$ and $[\delta ,m_a]=m_{\delta (a)}$, which is the Leibniz rule. Thus $\delta$ is a derivation of an algebra. Conversely any derivation satisfies this commutator identity. The sum is direct because a derivation that is multiplication must vanish at $1$.