Transvections as generators of the special linear group over the quaternions (Q1115976)

Transvections play a prominent role as generators of the special linear group of a finite dimensional vector space over a commutative field. A naturally arising question is to determine the minimal number \(\ell (M)\) of transvections needed to factorize each transformation M in the special linear group into transvections. The authors tackle the corresponding problem for a right vector space over a special skewfield, namely the quaternions. Using normal forms they show \(\ell (M)=\dim (M-1)V\) for a large class of transformations M. For all other transformations it is easy to see that \(\ell (M)=\dim (M-1)V+i\) where \(0\leq i\leq 3\).