Transvections as generators of the special linear group over the quaternions

Abstract

Let V be a finite-dimensional right vector space over the quaternions H . Each transformation M in the special linear group of V is a product of transvections T i , i.e. M = T 1… T t . The smallest t is called the length of M, t = l( M). We show that l( M) = dim B( M) ≥3, M≠3 J 1( λ) where λ ϵ C\\ R , and M μ is neither simple nor the identity for any μ ϵ R . In any case l( M) = dim B( M) + i, where 0≤ i≤3.