Split Octonions and Triality in (4+4)-Space

The known equivalence of 8-dimensional chiral spinors and vectors is discussed for (4+4)-space within the context of the algebra of the split octonions. It is shown that the complete algebra of hyper-complex octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements of the split octonions. Trilinear form, which is invariant under SO(4,4) transformations for vectors and corresponding Spin(4,4) transformations for spinors, is explicitly written using both purely matrix and purely octonionic representations.