ABSTRACT

Multiplications by unit complex numbers or quaternions generate the respective orthogonal groups SO2 and SO4. This chapter proves that the corresponding result that the multiplications by unit octonions generate SO8. Since the associative law fails for octonions, it is no longer true that the product of two multiplications of the same kind is another. The chapter shows that in fact the octonion multiplications of either kind generate the whole group SO8. Since positive scalar factors can be absorbed into all three types of multiplication, we need not restrict to units: Lx can be a product of six right multiplications only if x is real. The chapter explores the scenario for products of multiplications that may be of mixed types.