@pschwahn - Oh my god, how dumb I am! span{i,j,ℓ} is definitely invariant because I'm rotating those vectors into linear combinations of themselves. So... back to the drawing board.
By the way, it was not completely trivial to check that if we rotate i, j, ℓ we get a new basic triple, because the concept of basic triple is defined using octonion multiplication. So when I succeeded, I thought I was being clever. I seem to be getting a bunch of SO(3) subgroups of G₂ (each basic triple gives me one), but none that act irreducibly on the imaginary octonions.
I suspect that a 'generic' SO(3) subgroup of G₂ will act irreducibly. Sometimes 'generic' things are the ones for which there's no simple formula. But I still hope there's something nice going on here. I want to connect 3d geometry to 7d or 8d geometry in a nice way.
There's a nice way to build the octonions from ℝ³ where you take the exterior algebra Λ(ℝ³), which is 8-dimensional, and give it a multiplication. SO(3) acts on Λ(ℝ³) in the usual way, and this preserves the octonion multiplication, but unfortunately it acts reducibly since it preserves each grade Λⁱ(ℝ³).