Real and Unreal

template<typename T> T  real(octonion<T> const & o);
template<typename T> octonion<T> unreal(octonion<T> const & o);

These return o.real() and o.unreal() respectively.

conj

template<typename T> octonion<T> conj(octonion<T> const & o);

This returns the conjugate of the octonion.

sup

template<typename T> T   sup(octonion<T> const & o);

This return the sup norm (the greatest among abs(o.R_component_1())...abs(o.R_component_8())) of the octonion.

l1

template<typename T> T   l1(octonion<T> const & o);

This return the l1 norm (abs(o.R_component_1())+...+abs(o.R_component_8())) of the octonion.

abs

template<typename T> T   abs(octonion<T> const & o);

This return the magnitude (Euclidean norm) of the octonion.

norm

template<typename T> T  norm(octonion<T>const  & o);

This return the (Cayley) norm of the octonion. The term "norm" might be confusing, as most people associate it with the Euclidean norm (and quadratic functionals). For this version of (the mathematical objects known as) octonions, the Euclidean norm (also known as magnitude) is the square root of the Cayley norm.