have a look at these in memory so you stay sane.
v[0,0.2,4] 3D vector
e₄[0,0,0,1]
bv₃₄[0.3,0.8] thats a grade 2
eₙ these always square to the signature
eₘ eₙ these are always orthonormal so will square to -1.
beware - (e₂ ∧ e₁)² will not square to 1.
1st because: (e₂ ∧ e₁)² = (-1 (e₁ ∧ e₂))² = (-1)² * (e1 ∧ e₂)²
and that is 1 * -1 so still -1.
2nd because: (e₂ ∧ e₁)² = (e₂e₁)² = (e₂e₁)(e₂e₁) = -(e₂e₂)(e₁e₁) = -1 * 1 = -1
3rd: multiplying by e₁∧e₂ means rotate 90° so no matter what you end up at 180° = -1
this seems not much but from that you can build!