Trivial pair of Lie ring actions is compatible

Statement
Suppose $$M$$ and $$N$$ are Lie rings. Suppose $$\alpha:M \to \operatorname{Der}(N)$$ and $$\beta:N \to \operatorname{Der}(M)$$ are both trivial homomorphisms. Then, $$\alpha$$ and $$\beta$$ are compatible with each other.

Related facts

 * Trivial pair of actions is compatible