Involutive automorphisms and derivations of the quaternions

Let (Formula presented) 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 (Formula presented) Der (H), we show that the matrix D G M₃((Formula presented)) that represents d on the linear subspace H0 ~ (Formula presented) of pure quaternions provides a pair of idempotent matrices AdjD and -D² that correspond bijectively to the involutary matrix Σ of a quaternion involution σ 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.