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
$\omega\in\Lambda^3 V^\*$



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> **Question.** What is a coordinate-free description of a $G\_2$-invariant alternating trilinear form on $V$?
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Coordinate-free description of an alternating trilinear form on pure octonions
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).


