On split Lie algebras with symmetric root systems

We develop techniques of connections of roots for split Lie algebras with symmetric root systems. We show that any of such algebras L is of the form L = u + Sigma(j) I-j with u a subspace of the abelian Lie algebra H and any I-j a well described ideal of L, satisfying [I-j, I-k] = 0 if j not equal k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected.