Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in . As a consequence of the answers to the above question, we offer a characterization of the real elements in .
Suppose is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group . A non-degenerate S-invariant subspace of is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in . The level of a unipotent T is the least integer k such that (T-I)k=0. We also classify the levels of unipotents in .