If \(L\) is a subalgebra of a Lie algebra \(Q\); and if given \(p; q \in Q\) with \(p \not= 0\); there exists \(x \in L\) such that \([x; p] \not= 0\) and \([x; {}_L (q)] \subseteq L\) for \({}_L (q)\) the linear span in \(Q\) of \(q\) and the elements \(\text{ad}x_1 \cdots \text{ad}x_n q\) for \(x_1, \dots, x_n \in L\); then \(Q\) is called an algebra of quotients of \(L\). The author shows that if \(L\) is semiprime, prime, or nondegenerate, then \(Q\) is semiprime, prime, or nondegenerate, respectively. She also constructs the maximal algebra of quotients of every semiprime Lie algebra and shows that every semisimple Lie algebra is its own maximal algebra of quotients.