I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e\_1,e\_2,e\_3$, $v\_1,v\_2,v\_3$ be two triples of mutually orthogonal vectors. 


**Is there a (unique?) transformation in $Spin(7)$ that takes $e\_i$ to $v\_i$, $i=1,2,3$?** 


If so, which is the easiest way to convince oneself? If not, why? And what about quadruples of vectors?


Thank you very much for you attention and answers.



Transitivity of $Spin(7)$ in triples of vectors
We know that $\operatorname{Spin}(7)$ acts transitively on unit vectors, with stabilizer $G\_2$, so we need only prove that $G\_2$ acts transitively on pairs of orthogonal unit vectors. But then apply the same argument, since $G\_2$ acts transitively on the unit sphere in $\mathbb{R}^7$, with stabilizer $SU(3)$, which also acts transitively on the unit sphere in $\mathbb{R}^6$. So $\operatorname{Spin}(7)$ acts transitively on triples of mutually orthogonal unit vectors. But the dimension count ensures that it is not unique, apparently, following Semyon Alesker's argument.


