Suppose that formula_1 is a semisimple Lie algebra over the field of real numbers formula_26. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries . By Sylvester's law of inertia, the number of positive entries is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis, and is called the index of the Lie algebra formula_1. This is a number between and the dimension of formula_1 which is an important invariant of the real Lie algebra. In particular, a real Lie algebra formula_1 is called compact if the Killing form is negative definite (or negative semidefinite if the Lie algebra is not semisimple). Note that this is one of two inequivalent definitions commonly used for compactness of a Lie algebra; the other states that a Lie algebra is compact if it corresponds to a compact Lie group. The definition of compactness in terms of negative definiteness of the Killing form is more restrictive, since using this definition it can be shown that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups.