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 formula_1 such that the structure constants with all upper indices are completely antisymmetric.