Structure of isometry group of bilinear spaces

We describe the structure of the isometry group G of a finite-dimensional bilinear space over an algebraically closed field of characteristic not two. If the space has no indecomposable degenerate orthogonal summands of odd dimension, it admits a canonical orthogonal decomposition into primary components and G is isomorphic to the direct product of the isometry groups of the primary components. Each of the latter groups is shown to be isomorphic to the centralizer in some classical group of a nilpotent element in the Lie algebra of that group. In the general case, the description of G is more complicated. We show that G is a semidirect product of a normal unipotent subgroup K with another subgroup which, in its turn, is a direct product of a group of the type described in the previous paragraph and another group H which we can describe explicitly. The group H has a Levi decomposition whose Levi factor is a direct product of several general linear groups of various degrees. We obtain simple formulae for the dimensions of H and K.