  We present an alternative account of the problem of classifying and finding
normal forms for arbitrary bilinear forms. Beginning from basic results
developed by Riehm, our solution to this problem hinges on the classification
of indecomposable forms and in how uniquely they fit together to produce all
other forms. We emphasize the use of split forms, i.e., those bilinear forms
such that the minimal polynomial of the asymmetry of their non-degenerate part
splits over ground field, rather than restricting the field to be algebraically
closed. In order to obtain the most explicit results, without resorting to the
classification of hermitian, symmetric and quadratic forms, we merely require
that the underlying field be quadratically closed.
