Let $k$ be a field, let $\mathfrak g$ be a finite-dimensional [Lie algebra](/page/Lie%20Algebra) over $k$, and let $(e_1,\dots,e_n)$ be an ordered $k$-basis of $\mathfrak g$. Let $U(\mathfrak g)$ denote the universal enveloping algebra of $\mathfrak g$, and write the image of each $e_i$ in $U(\mathfrak g)$ again as $e_i$. Then the ordered monomials

\begin{align*} e_1^{a_1}e_2^{a_2}\cdots e_n^{a_n}, \end{align*}

where $a_1,\dots,a_n\in\mathbb Z_{\geq 0}$, form a $k$-basis of the [vector space](/page/Vector%20Space) $U(\mathfrak g)$.