# Coordinate-free description of an alternating trilinear form on pure octonions

Let $$O$$ denote the division algebra of octonions over $$\Bbb R$$, and write $$V$$ for the 7-dimensional quotient space $$O/{\Bbb R}$$. The compact group $$G_2:={\rm Aut}(O)$$ naturally acts on $$V$$, and clearly the 7-dimensional representation of $$G_2$$ in $$V$$ is isomorphic to its representation in the space of pure octonions.

I know from a classification of alternating trilinear forms in dimension 7 that there exists a $$G_2$$-invariant alternating trilinear form $$\omega\in\Lambda^3 V^*$$

Question. What is a coordinate-free description of a $$G_2$$-invariant alternating trilinear form on $$V$$?

• What about $(x,y,z)\mapsto\mathrm{Re}(x(yz)+y(zx)+z(xy)-x(zy)-y(xz)-z(yx))$? It's clearly invariant alternating. On $(i,j,k)$ its value is $-6$, so it's nonzero.
– YCor
Nov 30, 2020 at 15:20
• @YCor: Yes, it is clearly invariant and alternating. How did you compute the value on $(i,j,k)$? Nov 30, 2020 at 15:27
• Just from the table: $i(jk)=j(ki)=k(ij)=-i(kj)=-j(ik)=-k(ji)=-1$...
– YCor
Nov 30, 2020 at 15:40
• @YCor: Thank you, that is an answer! Nov 30, 2020 at 15:46

The form $$(x,y,z)\mapsto \mathrm{Re}(x(yz)+y(zx)+z(xy)−x(zy)−y(xz)−z(yx))$$ is clearly invariant and alternating. It is nonzero, since its value at $$(i,j,k)$$ (which satisfy the quaternions relations) is $$-6$$.

Actually, it can be checked that the symmetrized form $$\mathrm{Re}(x(yz)+y(zx)+z(xy)+x(zy)+y(xz)+z(yx))$$ vanishes. So the invariant form $$(x,y,z)\mapsto\mathrm{Re}(x(yz)+y(zx)+z(xy)$$ is already alternating (and takes the value $$-3$$ at $$(i,j,k)$$: it's actually zero modulo $$3$$ on the basis).

• How did you check that the symmetrized form vanishes? Nov 30, 2020 at 16:17
• I did a little brute Sage program this time to check all cases (to check the vanishing on all 343 possible triples).
– YCor
Nov 30, 2020 at 16:37
• Excellent! Thank you! Nov 30, 2020 at 16:43
• Also if I'm correct, the 3rd symmetric power, of dimension 84, decomposes in irreducibles as 77+7. In particular it doesn't contain the trivial representation, so there's no nonzero invariant symmetric trilinear form. (While the 3rd alternating power, of dimension 35, decomposes as 27+7+1.)
– YCor
Nov 30, 2020 at 22:27
• My calculation using Table 5 in the book by Onishchik and Vinberg shows that $V\otimes S^2 V$ does not contain the trivial representation. Thus indeed there is nonzero invariant symmetric bilinear form. Dec 1, 2020 at 10:38