Corollary 6

Suppose that the $K$-module $g$ is the direct sum of subalgebras $g_1, g_2, \ldots, g_n$ and that each $g_i$ is a free $K$-module. Let $U_i$ be the enveloping algebra of $g_i$ ($1 \leq i \leq n$). Let $\phi$ be the $K$-linear mapping of the $K$-module $U_1 \otimes_K \cdots \otimes_K U_n$ into $U$ defined by the multilinear mapping $(u_1, \ldots, u_n) \mapsto u_1 \ldots u_n$ of $U_1 \times \cdots \times U_n$ into $U$. Then $\phi$ is a $K$-module isomorphism.

Let $(x^i_{\lambda})_{\lambda \in L_i}$ be a basis of $g_i$. We totally order $L_1 \cup \cdots \cup L_n$ so that every element of $L_i$ exceeds every element of $L_j$ for $i \geq j$. Then the elements:

where $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_p \leq \cdots \leq v_1 \leq v_2 \leq \cdots \leq v_q$, constitute a basis of $U_1 \otimes_K \cdots \otimes_K U_n$. They are mapped by $\phi$ to the elements:

which constitute a basis of $U$. Hence the corollary.