Corollary 3

If $g$ admits a totally ordered basis $(x_\lambda)_{\lambda \in \Lambda}$, the elements $x_{\lambda_1} x_{\lambda_2} \ldots x_{\lambda_n}$ of the enveloping algebra $U$, where $(\lambda_1, \ldots, \lambda_n)$ is an arbitrary increasing finite sequence of elements of $\Lambda$, form a basis of the K-module $U$.

Let $\Lambda_n$ be the set of increasing sequences of $n$ elements of $\Lambda$. For $M = (\lambda_1, \ldots, \lambda_n) \in \Lambda_n$, let $y_M = x_{\lambda_1} \otimes x_{\lambda_2} \otimes \cdots \otimes x_{\lambda_n}$. Let $W$ be the submodule of $T^n$ with basis $(y_M)_{M \in \Lambda_n}$. Corollary 1 shows that the restriction of $\psi_n$ to $W$ is an isomorphism of $W$ onto a supplement of $U_{n-1}$ in $U_n$. But

whence the corollary.