  In this paper we examine how the notion of algebra of quotients for Lie
algebras ties up with the corresponding well-known concept in the associative
case. Specifically, we completely characterize when a Lie algebra $Q$ is an
algebra of quotients of a Lie algebra $L$ in terms of the associative algebras
generated by the adjoint operators of $L$ and $Q$ respectively. In a converse
direction, we also provide with new examples of algebras of quotients of Lie
algebras and these come from associative algebras of quotients. In the course
of our analysis, we make use of the notions of density and multiplicative
semiprimeness to link our results with the maximal symmetric ring of quotients.
