What is "split-octonion"

In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split-signature (4,4) whereas the octonions have a positive-definite signature (8,0).

Up to isomorphism, the octonions and the split-octonions are the only two octonion algebras over the real numbers. There are corresponding split octonion algebras over any field F.