C08.2 · Definitions/conventions

For native Zorn coordinates \(z=(\alpha,u,v,\beta)\), define \[ z z'= (\alpha\alpha'+u\cdot v',\ \alpha u'+\beta' u+v\times v',\ \alpha'v+\beta v'-u\times u',\ v\cdot u'+\beta\beta'). \] This is the same product used in the preceding foundation articles after their explicitly stated coordinate change. No table is recalled or inserted.

Use the trace-zero basis \[ s=\frac1{\sqrt2}(1,0,0,-1),\qquad u_i=(0,\epsilon_i,0,0),\qquad v_i=(0,0,\epsilon_i,0), \] where \(\epsilon_i\) is the standard real vector basis. Let \([x,y]=xy-yx\).

On this space define the reflected two-parameter symmetric form \[ B_{a,b}(s,s)=b,\qquad B_{a,b}(u_i,v_j)=B_{a,b}(v_j,u_i)=a\delta_{ij}, \] with all other basic pairings zero. Its required compatibility is \[ B_{a,b}([x,y],z)=B_{a,b}(x,[y,z]). \] This is the source's Frobenius compatibility condition on the commutator. It is **not** a claim that the octonion product is associative or that its commutator is a Lie bracket.

The reflected trace-weight class is stipulated as in LIB2-053: \[ \tau_{a,b}=b\,\operatorname{Tr}_{1} +a\,\operatorname{Tr}_{U}+a\,\operatorname{Tr}_{V}, \qquad r=\frac{\tau_{a,b}(I)}{a\,\dim U}. \] The two three-dimensional summands have a common weight because this is the registered reflected class. In the real Zorn realization, \(U,V\) are real dual subspaces under the usual determinant-preserving linear action on the vector coordinates. The source's notation \(1\oplus3\oplus\bar3\) is retained as its module bookkeeping; this construction does **not** embed a compact colour group in split \(G_2\).

For the separate signed-channel test, use the imaginary subspace of the article's split-octonion algebra and its norm pairing. In the balanced basis \((e_1,e_2,e_3,f_1,f_2,f_3,\ell)\), each vector is norm-orthonormal with sign \(\eta_a\). For a subset \(S\) of this fixed channel basis, define \[ R(S)=\sum_{a\in S}\eta_a. \] That specific unit-weight contraction is the object tested by LIB2-107. Arbitrary channel amplitudes are not included in this definition.