This is demonstrated below in most of the non-trivial cases. When the form is symmetric, is called . When it is skew-symmetric then is called . This applies to the real and the complex cases. The quaternionic case is empty since no nonzero bilinear forms exists on quaternionic vector spaces. The real case breaks up into two cases, the symmetric and the antisymmetric forms that should be treated separately. If is symmetric and the vector space is real, a basis may be chosen so that The number of plus and minus-signs are independent of the particular basis. In the case one writes where is the number of plus signs and is the number of minus-signs, . If the notation is . The matrix is in this case after reordering the basis if necessary. The adjoint operation () then becomes which reduces to the usual transpose when or is 0. The Lie algebra is found using equation () and a suitable ansatz (this is detailed for the case of below), and the group according to () is given by The groups and are isomorphic through the map For example, the Lie algebra of the Lorentz group could be written as Naturally, it is possible to rearrange so that the -block is the upper left (or any other block). Here the "time component" end up as the fourth coordinate in a physical interpretation, and not the first as may be more common. If is skew-symmetric and the vector space is real, there is a basis giving where . For one writes In case one writes or . From the normal form one reads off By making the ansatz where are -dimensional matrices and considering (), one finds the Lie algebra of ,  and the group is given by  Like in the real case, there are two cases, the symmetric and the antisymmetric case that each yield a family of classical groups. If case is symmetric and the vector space is complex, a basis with only plus-signs can be used. The automorphism group is in the case of called . The lie algebra is simply a special case of that for , and the group is given by  In terms of classification of simple Lie algebras, the are split into two classes, those with odd with root system and even with root system . For skew-symmetric and the vector space complex, the same formula, applies as in the real case. For one writes In case one writes or . The Lie algebra parallels that of , and the group is given by  In the sequilinear case, one makes a slightly different approach for the form in terms of a basis, The other expressions that get modified are The real case, of course, provides nothing new.