3 Foundations: G2 Cross Product
The 7-dimensional cross product is intimately connected to octonion multiplication and defines the \(G_2\) holonomy structure.
3.1 The Fano Plane
The Fano plane has 7 lines (cyclic triples):
The Fano plane has exactly 7 lines.
The structure constants \(\varepsilon _{ijk}\) for the 7D cross product:
\(\varepsilon _{ijk} = +1\) for \((i,j,k)\) a cyclic permutation of a Fano line
\(\varepsilon _{ijk} = -1\) for anticyclic permutations
\(\varepsilon _{ijk} = 0\) otherwise
3.2 Cross Product Definition
For \(u, v \in \mathbb {R}^7\), the cross product is:
For all \(i, j, k\): \(\varepsilon _{ijk} = -\varepsilon _{jik}\)
3.3 Cross Product Properties
The cross product is bilinear:
\(u \times v = -v \times u\)
Follows from \(\varepsilon _{ijk} = -\varepsilon _{jik}\) and sum reindexing.
\(u \times u = 0\)
3.4 Lagrange Identity (B4)
\(\displaystyle \sum _k \varepsilon _{ijk} \varepsilon _{lmk}\)
The 7D correction to the Kronecker formula:
\(\psi _{ijlm} = -\psi _{ljim}\) (verified for all \(7^4 = 2401\) index combinations)
\(\displaystyle \sum _{i,j,l,m} \psi _{ijlm} \, u_i u_l v_j v_m = 0\)
Antisymmetric tensor \(\psi \) contracted with symmetric \(u_i u_l\) vanishes.
Expand \(\| u \times v\| ^2\) via coordinate sums. The \(\varepsilon \)-contraction decomposes into Kronecker deltas plus \(\psi _{ijlm}\) terms. By antisymmetry of \(\psi \) (verified for all 2401 cases), the \(\psi \)-terms vanish under symmetric contraction \(u_i u_l v_j v_m\). The Kronecker terms yield \(\| u\| ^2 \| v\| ^2 - \langle u,v \rangle ^2\).