If are two ideals in a Lie algebra with zero intersection, then and are orthogonal subspaces with respect to the Killing form. The orthogonal complement with respect to of an ideal is again an ideal. If a given Lie algebra is a direct sum of its ideals , then the Killing form of is the direct sum of the Killing forms of the individual summands. Matrix elements Given a basis of the Lie algebra , the matrix elements of the Killing form are given by Here in Einstein summation notation, where the are the structure coefficients of the Lie algebra. The index functions as column index and the index as row index in the matrix . Taking the trace amounts to putting and summing, and so we can write The Killing form is the simplest 2-tensor that can be formed from the structure constants. The form itself is then In the above indexed definition, we are careful to distinguish upper and lower indices (co- and contra-variant indices). This is because, in many cases, the Killing form can be used as a metric tensor on a manifold, in which case the distinction becomes an important one for the transformation properties of tensors. When the Lie algebra is semisimple over a zero-characteristic field, its Killing form is nondegenerate, and hence can be used as a metric tensor to raise and lower indexes. In this case, it is always possible to choose a basis for such that the structure constants with all upper indices are completely antisymmetric. The Killing form for some Lie algebras are (for in viewed in their fundamental matrix representation): The table shows that the Dynkin index for the adjoint representation is equal to twice the dual Coxeter number. Connection with real forms Suppose that is a semisimple Lie algebra over the field of real numbers . 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 . This is a number between and the dimension of which is an important invariant of the real Lie algebra. In particular, a real Lie algebra is called compact if the Killing form is