A Method to Find the Solution of the Linear Octonionic Equation α(xα) = (αx)α = αxα = ρ

Because of the non-commutativity of quaternions and octonions, it is rather diffucult to find an explicit method for the solution of the equation with one unknown such that α(xα) = (αx)α = αxα = ρ in quaternions and octonions. In this note, firstly we consider the linear octonionic equation with one unknown of the form: α(xα) = (αx)α = αxα = ρ, with0 6= α ∈ O secondly present a method, which is reduce this octonionic equation to an equation with the left and right coefficients to a real system of eight equations to find the solutions of this equation, and finally reach the solution of the linear octonionic equation from this real system. We end the paper with one example to illustrate the results of this paper.