  The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie
algebra and to prove some elementary properties of $\operatorname{D}$-Lie
algebras, the category of $\operatorname{D}$-Lie algebras, the category of
modules on a $\operatorname{D}$-Lie algebra and extensions of
$\operatorname{D}$-Lie algebras. A $\operatorname{D}$-Lie algebra is an
$A/k$-Lie-Rinehart algebra equipped with an $A\otimes_k A$-module structure and
a canonical central element $D$ and a compatibility property between the
$k$-Lie algebra structure and the $A\otimes_k A$-module structure. Several
authors have studied non-abelian extensions of Lie algebras, super Lie
algebras, Lie algebroids and holomorphic Lie algebroids and we give in this
note an explicit constructions of all non-abelian extensions a
$\operatorname{D}$-Lie algebra $\tilde{L}$ by an $A$-Lie algebra $(W,[,])$
where $\tilde{L}$ is projective as left $A$-module and $W$ is an $A\otimes_k
A$-module with $IW=0$ for $I$ the kernel of the multiplication map. As a
corollary we get an explicit construction of all non-abelian extensions of an
$A/k$-Lie-Rinehart algebra $(L,\alpha)$ by an $A$-Lie algebra $(W,[,])$ where
$L$ is projective as left $A$-module.
