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.