Upon visiting Prof. Nurowski's homepage (<http://www.fuw.edu.pl/~nurowski/>), at the top of the page, there is the following $7$-dimensional cross product:


$e\_1 e\_2 = e\_4$, $e\_2 e\_3 = e\_5$,...


and so on, proceeding cyclically modulo $7$. My question is as follows. What is the stabilizer of this cross-product in $GL(7,\mathbb{R})$? Is it the compact $G\_2$ or some other real form of the $G\_2^{\mathbb{C}}$? I thought this question could be fun to post, particularly since this cross-product is particularly easy to memorize (I think I know the answer though).



What is the stabilizer of the following $7$-dimensional cross-product?
It was a "trivial" question. There is an element $g$ in $GL(7,\mathbb{R})$ which takes this $7$-dimensional cross-product into the standard one, related to the octonions. Thus the stabilizer of this cross-product is conjugate to the compact $G\_2$ inside $GL(7,\mathbb{R})$, via $g$, and is thus isomorphic to the compact $G\_2$. 


