Involutive automorphisms and derivations of the quaternions

Abstract

Let Q = (a,b/R) denote the quaternion algebra over the reals which is by the Frobenius Theorem either split or the division algebra H of Hamilton's quaternions. We have presented explicitly in [4] the matrix of a typical derivation of Q. Given a derivation d is an element of Der(H), we show that the matrix D is an element of M-3(R) that represents d on the linear subspace H-0 similar or equal to R-3 of pure quaternions provides a pair of idempotent matrices AdjD and - D-2 that correspond bijectively to the involutary matrix Sigma of a quaternion involution sigma and present several equations involving these matrices. In particular, we deal with commuting derivations of H and introduce some results to guarantee commutativity. We also mention briefly eigenspace decomposition of a derivation.