Let \(L\) be a finite dimensional Lie algebra over a field. The Frattini subalgebra, \(F(L)\), of \(L\) is the intersection of the maximal subalgebras of \(L\); the Frattini ideal, \(\varphi(L)\), of \(L\) is then the largest ideal of \(L\) contained in \(F(L)\). We say that \(L\) is elementary if \(\varphi(B)= 0\) for all subalgebras \(B\) of \(L\). This is analogous to the definition of an elementary group given by \textit{H. Bechtell} [Trans. Am. Math. Soc. 114, 355--362 (1965; Zbl 0142.26004)]. A Lie algebra \(L\) over a field of characteristic zero is elementary if and only if every subalgebra of \(L\) is complemented (in the sense of \textit{B. Kolman} [J. Sci. Hiroshima Univ., Ser. A-I 31, 1--11 (1967; Zbl 0166.04004)]). This paper studies elementary Lie algebras, and gives their structure quite explicitly when the ground field is restricted to be algebraically closed of characteristic zero. The development follows that of the corresponding group-theoretic concept, though the techniques of proof are quite different. Also studied here are Lie algebras \(L\) such that \(L/\varphi(L)\) is elementary. These have also been studied by \textit{E. L. Stitzinger} [Pac. J. Math. 34, No. 1, 177--182 (1970; Zbl 0203.33801)]. Finally, it is shown that any Lie algebra \(L\) has a unique ideal \(E(L)\) which has the property that \(E(L)\) is contained in any ideal \(B\) of \(L\) for which \(L/B\) is elementary, and some of the properties of \(E(L)\) are investigated.