Classical group

The classical groups can uniformly be characterized in a different way using real forms. The classical groups (here with the determinant 1 condition, but this is not necessary) are the following:

A classical group is a group that preserves a bilinear or sesquilinear form on finite-dimensional vector spaces over

The real case breaks up into two cases, the symmetric and the antisymmetric forms that should be treated separately.

Like in the real case, there are two cases, the symmetric and the antisymmetric case that each yield a family of classical groups.

In the sequilinear case, one makes a slightly different approach for the form in terms of a basis,

The real case, of course, provides nothing new. The complex and the quaternionic case will be considered below.

When dealing with quaternionic groups it is convenient to represent quaternions using complex 2×2-matrices,