On linear equations over split-octonions

Abstract

Over an algebraically closed field, we describe the affine varieties of solutions to the linear equations (ax)b=c and a(bx)=c over the split-octonions. We also determine the dimensions of the solution sets of arbitrary linear monomial equations in the split-octonions. Moreover, we show that if a linear monomial equation over the split-octonions with nonzero constant term has at least two solutions, then it necessarily possesses an invertible solution.