An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify the reversible and strongly reversible elements in the quaternionic special linear group $ \mathrm {SL}(n,\mathbb {H})$ and quaternionic projective linear group $ \mathrm {PSL}(n,\mathbb {H})$. We prove that an element of $ \mathrm {SL}(n,\mathbb {H})$ (resp. $ \mathrm {PSL}(n,\mathbb {H})$) is reversible if and only if it is a product of two skew-involutions (resp. involutions).

An element g in a group G is called reversible if g is conjugate to g−1 in G. An element g in G is strongly reversible if g is conjugate to g−1 by an involution in G. The group of affine transformations of $\mathbb D^n$ may be identified with the semi-direct product $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $, where $\mathbb D:=\mathbb R, \mathbb C$ or $ \mathbb H $. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{GL}(n, \mathbb D) \ltimes \mathbb D^n $.