C03.1 · Definitions/conventions

Write the split Albert element as \[ X=\begin{pmatrix}d_1&z&\bar y\\\bar z&d_2&x\\y&\bar x&d_3\end{pmatrix}, \qquad N(X)=d_1d_2d_3-d_1n(x)-d_2n(y)-d_3n(z) +2\operatorname{Re}((zx)y). \] The eight coordinates of each octonion use the quaternion-half Cayley–Dickson product of C01.1. All products here are bilinear binary products with the parentheses displayed. Let \[ T(X,Y)=\operatorname{Tr}(X\circ Y),\qquad T(X^\#,Y)=dN_X(Y). \] In the coordinate order \((d_1,d_2,d_3,z,y,x)\), the trace matrix is \(D=\operatorname{diag}(1,1,1,2\eta_8,2\eta_8,2\eta_8)\), with \(\eta_8=\operatorname{diag}(1,1,1,1,-1,-1,-1,-1)\). Its indefiniteness does not make the alternating form below degenerate.

Use coordinates \((a,X;b,Y)\) on \(\mathcal F=\mathbb R\oplus J\oplus\mathbb R\oplus J\), and define \[ \Omega((a,X;b,Y),(c,U;d,V))=ad-bc+T(X,V)-T(Y,U). \] The canonical covector is \(p=(b,DY)\), not the raw coordinate vector \((b,Y)\). With \(q=(a,X)\), the symplectic matrix is the canonical block matrix. This distinction fixes all Hamiltonian signs used below: \(\dot q=\partial_p H\), \(\dot p=-\partial_q H\).

Our quartic convention is \[ I_4=-(ab-T(X,Y))^2-4aN(X)-4bN(Y)+4T(X^\#,Y^\#). \] It is one half of the quartic \(q\) in [K04] §3.1, after the displayed coordinate matching. That normalization comparison is explicit; it is not a claim that all published Freudenthal conventions have these signs.