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We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals) over a field of characteristic distinct from 2. There is an associative bilinear form on such a Lie algebra; we study its connections with the Killing form. Any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal numbers of extremal generators for the Lie algebras of type A n n ≥ 1 , B n n ≥ 3 , C n n ≥ 2 , D n n ≥ 4 , E n n = 6 7 8 , F 4 and G 2 are shown to be n + 1, n + 1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.

We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to O(n)-adjoint orbits in o(n).