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A (vector space) basis B of a Lie algebra is said to be very nilpotent if allthe iterated brackets of elements of B are nilpotent. In this note, we prove arefinement of Engel's Theorem. We show that a Lie algebra has a very nilpotentbasis if and only if it is a nilpotent Lie algebra. When g is a semisimple Liealgebra, this allows us to define an ideal of S((g^n)^*)^G whose associatedalgebraic set in g^n is the set of n-tuples lying in a same Borel subalgebra.

Very nilpotent basis and n-tuples in Borel subalgebras