My first thought was Atiyah & Macdonald's 'An Introduction to Commutative Algebra' - which has already been mentioned - and 'anything by J.P. Serre' (that's old enough, of course!). It appears that not quite everything in this latter category has been mentioned; notably, 'Algebres de Lie Semi-simple Complexes', first published in 1966. There is also a later English translation, 'Complex Semisimple Lie Algebras' published in 1987.

While not quite an introduction, I find myself referring back to this text often for its streamlined, beautiful exposition (a hallmark of Serre). It also has the best exposition of root systems I've encountered.

Furthermore, another classic text on semisimple Lie algebras (J. Humphreys - 'Introduction to Lie Algebras & Representation Theory') is a 'fleshing out' of Serre's notes. Actually, Humphreys's textbook was first published in 1972 so might squeeze onto this list too?