Tensor product of Lie rings

Definition
Suppose $$M$$ and $$N$$ are Lie rings and $$\alpha:M \to \operatorname{Der}(N)$$ and $$\beta:N \to \operatorname{Der}(M)$$ is a compatible pair of actions of Lie rings. We define the tensor product $$M \otimes N$$ for this pair of actions as follows. It is the quotient of the free Lie ring on formal symbols of the form $$m \otimes n$$ ($$m \in M, n \in N$$) by the following relations:


 * Additive in $$M$$: $$(m_1 + m_2) \otimes n = (m_1 \otimes n) + (m_2 \otimes n) \ \forall \ m_1,m_2 \in M, n \in N$$. Note that if we are dealing with Lie algebras instead of Lie rings, we will replace additivity by linearity in $$M$$ with respect to the ground ring.
 * Additive in $$N$$: $$m \otimes (n_1 + n_2) = (m \otimes n_1) + (m \otimes n_2) \ \forall \ m \in M, n_1,n_2 \in N$$. Note that if we are dealing with Lie algebras instead of Lie rings, we will replace additivity by linearity in $$N$$ with respect to the ground ring.
 * Expanding a tensor product involving one Lie bracket:
 * $$[m_1,m_2] \otimes n = m_1 \otimes \alpha(m_2)(n) - m_2 \otimes \alpha(m_1)(n) \ \forall \ m_1,m_2 \in M, n \in N$$
 * $$m \otimes [n_1,n_2] = \beta(n_2)m \otimes n_1 - \beta(n_1)(m) \otimes n_2 \ \forall m \in M, n_1,n_2 \in N$$

If both the actions are rewritten using $$\cdot$$, this simplifies to:
 * $$[m_1,m_2] \otimes n = m_1 \otimes (m_2 \cdot n) - m_2 \otimes (m_1 \cdot n) \ \forall \ m_1,m_2 \in M, n \in N$$
 * $$m \otimes [n_1,n_2] = (n_2 \cdot m) \otimes n_1 - (n_1 \cdot m) \otimes n_2 \ \forall m \in M, n_1,n_2 \in N$$
 * Expanding a Lie bracket of two pure tensors:

$$[(m_1 \otimes n_1),(m_2 \otimes n_2)] = -(\beta(n_1)(m_1)) \otimes (\alpha(m_2)(n_2))$$

If both the actions are rewritten using $$\cdot$$, this becomes:

$$[(m_1 \otimes n_1),(m_2 \otimes n_2)] = -(n_1 \cdot m_1) \otimes (m_2 \cdot n_2)$$

Maps and constructions
For the statements in these facts, we will use the same notation as in the definition above: $$M$$ and $$N$$ are Lie rings with a compatible pair of actions of Lie rings $$\alpha:M \to \operatorname{Der}(N)$$ and $$\beta:N \to \operatorname{Der}(M)$$.