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The group Spin$\left( 7\right) \ $is the universal cover of $SO\left( 7\right) $ and is characterized as the subgroup of $SO\left( 8\right) $ consisting of the automorphisms of the triple cross product $X:\mathbb{R}% ^{8}\times \mathbb{R}^{8}\times \mathbb{R}^{8}\rightarrow \mathbb{R}^{8}$ (see for instance [SW]). In particular, Spin$\left( 7\right) $ acts canonically on the Grassmannian $G\left( 3,8\right) $ of oriented 3-dimensional subspaces of $\mathbb{R}^{8}=\mathbb{O}$. We would like to know the isotropy subgroup $K$ of this action at $1\wedge i\wedge j$.

What we know:

Theorem 9.3 in [SW] asserts that Spin$\left( 7\right) $ acts transitively on \begin{equation*} \{(u,v,w,x)\in \mathbb{R}^{8}\mid u,v,w,X(u,v,w),x\text{ are orthonormal}\}% \text{.} \end{equation*} This implies the following:

  • The action of Spin$\left( 7\right) $ on $G\left( 3,8\right) $ is transitive and so the dimension of $K$ must be $6$.

  • Given a positively oriented orthonormal basis $w_{1},w_{2},w_{3}$ of $% 1\wedge i\wedge j$, then there exist $g\in $ Spin$~\left( 7\right) $ satisfying $g\left( 1\right) =w_{1}$, $g\left( i\right) =w_{2}$ and $g\left( j\right) =w_{3}$ (we recall that $X\left( 1,i,j\right) =k$).

On the other hand, we know that the group $S^{3}\times S^{3}\subset G_{2}\subset $ Spin$\left( 7\right) $ acts on the octonions as follows: \begin{equation*} \left( u,v\right) \cdot \left( x,y\right) =\left( ux\bar{u},vy\bar{u}\right) \text{,} \end{equation*} where $\left( x,y\right) \in \mathbb{H}\times \mathbb{H}=\mathbb{O}$ (as sets). This gives us a 4-dimensional subgroup of $K$ given by \begin{equation*} \left\{ \left( e^{tk},v\right) \mid t\in \mathbb{R},v\in S^{3}\right\}, \end{equation*} since conjugation by $e^{tk}$ fixes 1 and rotates the oriented plane $ i\wedge j$ through the angle $2t$. In particular, if $w_{1}=1$, $w_{2}=\cos t~i+\sin t~j$ and $w_{3}=\cos t~j-\sin t~i$, then $g$ as above can be taken to be $g\left( x,y\right) =\left( e^{tk/2},1\right) \cdot \left( x,y\right) $ .

We would prefer to have the elements of $K$ explicit as linear isometries of $\mathbb{R}^{8},$ but, of course, it would be interesting to us to know the Lie algebra of $K$ as a subset of $o(8)$ or the isomorphism type of $K$.

[SW] D.A. Salamon, T. Walpuski, Notes on the octonions. Proceedings of the Gökova Geometry-Topology Conference 2016, 1--85, Gökova Geometry/Topology Conference (GGT), Gökova, 2017. Also: arXiv:1005.2820 [math.RA]

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