The 4D Cross Product in Split Space: Isoclinic Decomposition of SO(2,2) and Its Absence for SO(1,3)

The 4D cross product Im(ĀB) and its left invariance extend from SO(4) to SO(2,2) through split quaternions, giving R = L(h)(1⊕M) with M ∈ SO(1,2), Δ₁ = tr M + tr M′, the eigenvalue types, and a split-quaternion QR decomposition. The split cross product is the rotation axis of zero-split octonions. SO(1,3) has no isoclinic decomposition because ⋆² = −1 and boosts do not form a group.