A left (right) *action* of a Lie algebra ${\mathfrak{g}}$ on ${\mathcal{A}}$ is a Lie algebra homomorphism (antihomomorphism) ${\alpha}:{\mathfrak{g}}\to\operatorname{Der}({\mathcal{A}})$.

Assume that we are given a commutative algebra ${\mathcal{A}}$ and a Lie algebra ${\mathfrak{g}}$ which is also a faithful ${\mathcal{A}}$-module. The pair $({\mathcal{A}},{\mathfrak{g}})$ is a *Lie–Rinehart algebra* if there exists an ${\mathcal{A}}$-module morphism ${\alpha}:{\mathfrak{g}}\to\operatorname{Der}({\mathcal{A}})$, called the *anchor*, satisfying the compatibility condition $$a[X, Y] = [X, a\,Y] - {\alpha}(X)a\,Y,
\label{eq:compatibility}$$ for $a\in{\mathcal{A}}, X,Y\in {\mathfrak{g}}$. If we write $m_a$ for the multiplication operator $m_a(X)=a\,X$ and $\operatorname{ad}_X$, as usual, for the adjoint operator $\operatorname{ad}_X(Y)=[X, Y]$, the compatibility condition is rewritten as $$[\operatorname{ad}_X, m_a] = m_{{\alpha}(X)a}.$$ Often, in the definition of Lie–Rinehart algebra, the apparently stronger condition that the anchor be also a left action of ${\mathfrak{g}}$ on ${\mathcal{A}}$ is required. But, as it turns out, these two definitions are equivalent.

The concept essentially coincides with Kastler and Stora’s Lie–Cartan pairs [@GoodPair]. As already indicated $\bigl({\mathcal{F}}(M),{\mathfrak{X}}(M),{{\mathrm{id}}}\bigr)$ is a Lie–Rinehart algebra. A more general example of Lie–Rinehart algebra may be $\bigl({\mathcal{F}}(M),
{\Gamma}(M,E),{\alpha}\bigr)$, where ${\Gamma}(M,E)$ is the ${\mathcal{F}}(M)$-module of sections of a vector bundle $E$ over $M$, on which a Lie bracket and an anchor ${\alpha}$ (hence a vector bundle map $E\to TM$, denoted in the same way) 