Tensor product of Lie rings is commutative up to natural isomorphism

Statement
Suppose $$M$$ and $$N$$ are (not necessarily abelian) Lie rings with a compatible pair of Lie ring actions $$\alpha:M \to \operatorname{Der}(N)$$ and $$\beta: N \to \operatorname{Der}(M)$$ on each other. Consider the tensor products $$M \otimes N$$ and $$N \otimes M$$ corresponding to these actions. The tensor products $$M \otimes N$$ and $$N \otimes M$$ are naturally isomorphic, with the natural isomorphism $$M \otimes N \to N \otimes M$$ given as follows on a generating set:

$$m \otimes n \mapsto -(n \otimes m)$$

Related facts

 * Tensor product of groups is commutative up to natural isomorphism