C02.1 · Definitions/conventions

Use the split Cayley–Dickson product \[ (p,q)(r,s)=(pr+\bar s q,\;sp+q\bar r), \qquad \overline{(p,q)}=(\bar p,-q), \] with quaternion halves. The basis is \((1,e_1,e_2,e_3,f_1,f_2,f_3,\ell)\), with \(f_i=-e_i\ell\), as in C01.1. Its norm is positive on the first quaternion half and negative on the second.

An element is written \[ X= \begin{pmatrix} a&z&\bar y\\ \bar z&b&x\\ y&\bar x&c \end{pmatrix},\qquad a,b,c\in\mathbb R,\quad x,y,z\in\mathbb O_s . \] The code's coordinate order is \((a,b,c;\ z_0,\ldots,z_7;\ y_0,\ldots,y_7;\ x_0,\ldots,x_7)\). Define \[ X\circ Y=\frac{XY+YX}{2},\quad t(X)=a+b+c, \] \[ S(X)=ab+ac+bc-n(x)-n(y)-n(z), \] \[ N(X)=abc-a\,n(x)-b\,n(y)-c\,n(z) +2\operatorname{Re}((zx)y). \] The displayed parentheses are part of the definition. This cubic is not an unqualified determinant of an associative matrix algebra.

Set \[ X^\#=X\circ X-t(X)X+S(X)I. \] The full real linear map is \(L_A:Y\mapsto A\circ Y\). Its matrix columns are the actual products of \(A\) with the declared coordinate basis—not a list of eigenvalues encoded as a matrix.