are supposed given. If the fibres have dimension bigger than one, then a linear map satisfying  is not only automatically a Lie algebra morphism, but also a ${\mathcal{F}}(M)$-module morphism. These geometrical examples are called Lie algebroids. A Lie algebroid is called *transitive* when it is onto fibrewise; totally intransitive when ${\alpha}=0$. For examples of this, consider a principal bundle $P(M,G,\pi)$ over $M$; if $VP$ is the vertical bundle over $P$, we have the exact sequences of vector bundles $$0 \to VP   \hookrightarrow TP   \to TM \to 0 {\quad\mbox{and}\quad}
0 \to VP/G \hookrightarrow TP/G \to TM \to 0;$$ the second being essentially the Atiyah sequence; and then $(C^\infty(M),{\Gamma}(M,TP/G),T\pi/G)$ is a transitive Lie algebroid; while $\bigl(C^\infty(M),{\Gamma}(M,VP/G)\bigr)$ is totally intransitive.

Whenever we have a Lie–Rinehart algebra, we can algebraically define a differential calculus. For instance a $n$-form is a skewsymmetric $n$-linear map from ${\mathfrak{g}}$ to ${\mathcal{A}}$. If we define $d$ on 1-forms by $$d{\beta}(X,Y) = {\alpha}(X){\beta}(Y) - {\alpha}(Y){\beta}(X) - {\beta}\bigl([X, Y]\bigr),$$ then certainly $d$ can be extended so $d^2=0$. Also, let $V$ be an ${\mathcal{A}}$-module. A $V$-connection in the sense of [@GoodPair; @Bigotes2] is a linear assignment to each element $X\in {\mathfrak{g}}$ of a linear map $\rho(X):V\to V$ such that, for $v\in V$, 