Proposition 2

The homomorphism $\phi$ is an algebra isomorphism

The mapping $\sigma': (x_1, x_2) \mapsto \sigma_1(x_1) \otimes 1 + 1 \otimes \sigma_2(x_2)$ ($x_1 \in g_1, x_2 \in g_2$) is an $\alpha$-mapping of $g$ into $U_1 \otimes_K U_2$ and hence there exists (no. 1, Proposition 1) a unique homomorphism $\tau$ of $U$ into $U_1 \otimes_K U_2$ mapping 1 to 1, such that:

Now $\phi \circ \tau \circ \sigma = \phi \circ \sigma' = \sigma$ and $\tau \circ \phi \circ \sigma' = \tau \circ \sigma = \sigma'$, hence $\phi \circ \tau$ and $\tau \circ \phi$ are the identity mappings of $U$ and $U_1 \otimes_K U_2$ respectively. Hence the proposition.

$U_1 \otimes_K U_2$ is identified with $U$ under the isomorphism $\phi$. Then the canonical mapping of $g$ into $U$ is identified by (1) with the mapping:

Analogously, if $g_1, \ldots, g_n$ are Lie algebras over $K$ with enveloping algebras $U_1, \ldots, U_n$, the enveloping algebra $U$ of $g_1 \times \cdots \times g_n$ is canonically identified with $U_1 \otimes_K \cdots \otimes_K U_n$ and the canonical mapping of $g_1 \times \cdots \times g_n$ into $U$ is identified with the mapping:

($\sigma_i$ denoting the canonical mapping of $g_i$ into $U_i$).