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 ∈ Der ( H ) , we show that the matrix D ∈ M 3 ( R ) that represents d on the linear subspace H 0 ≃ R 3 of pure quaternions provides a pair of idempotent matrices AdjD and − D 2 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.