Classification of pencils of skew reflections on planes in the equiaffine space. (Q1179573)

Let \(k\) be a commutative field of characteristic distinct from 2 and \(A^ 3\) the 3-dimensional affine space over \(k\). If \(\sigma_ 1\), \(\sigma_ 2\), \(\sigma_ 3\) are members of the set \(S\) of involutory shears and \(\sigma_ 1\sigma_ 2\sigma_ 3\) is not in \(S\), then \(\sigma_ 4\) in \(S\) belongs to the pencil \(B(\sigma_ 1\sigma_ 2\sigma_ 3)\) if \(\sigma_ 1\sigma_ 2\sigma_ 3\sigma_ 4\), \(\sigma_ 2\sigma_ 1\sigma_ 3\sigma_ 4\), and \(\sigma_ 1\sigma_ 3\sigma_ 2\sigma_ 4\) are all products of fewer than four shears. The author shows that there are exactly three kinds of pencils.