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We present an alternative account of the problem of classifying and findingnormal forms for arbitrary bilinear forms. Beginning from basic resultsdeveloped by Riehm, our solution to this problem hinges on the classificationof indecomposable forms and in how uniquely they fit together to produce allother forms. We emphasize the use of split forms, i.e., those bilinear formssuch that the minimal polynomial of the asymmetry of their non-degenerate partsplits over ground field, rather than restricting the field to be algebraicallyclosed. In order to obtain the most explicit results, without resorting to theclassification of hermitian, symmetric and quadratic forms, we merely requirethat the underlying field be quadratically closed.

Equivalence and normal forms of bilinear forms