A general form of mappings preserving angle between vectors and their image in $R^8$ In $\mathbb R^8$, identified with the octonion algebra $\mathbb O$,  mappings $f: O \rightarrow \mathbb O$ of the form  $x \mapsto xu $ and  $  x \mapsto ux $, where $u$ is a fixed unit octonion (i.e. $\|u\|=1$), are orthogonal 
and preserving the angle between $x$ and $f(x)$ (i.e. $\frac{<f(x),x>}{\|x\|^2}=const_f$ for  $x\in \mathbb O$).
Is an arbitrary orthogonal mapping with this property of one of this form?
 A: All orthogonal mappings $\mathbb R^8 \to \mathbb R^8$ which have right angle between point and its image form 12-dimensional submanifold in $SO_8$. They can be seen as "complex structures" in $\mathbb R^8$ i.e. $O_8/U_4$. Let's denote this manifold as $\gimel$. It has two components. It can be defined as $\{x \in O_8:x^2=-1\}$.
Left and right multiplication by imaginary octonions form two copies $S^6 \subset \gimel$. General element in $\gimel$ is of the form $L_uL_{\bar v}$ or $R_uR_{\bar v}$ where $u,v$ are perpendicular unit octonions. In this presentation all $u,v$ lying on the circle $S^1 \subset S^7$ gives the same element $L_uL_{\bar v}$ or $R_uR_{\bar v}$.
This sentence proves that topologically manifold $\gimel$ is grassmanian $G_{2,6}^+$ (space of oriented planes in $\mathbb R^8$). We can use triality to map "left rotations" to "right rotations" and to "elementary rotations" in $SO_8$.
One may ask about $SO_{16}$. Set $\{x^2=-1\}$ is 14+12+...+2=56 dimensional there. We can try to express it in terms of Clifford algebra $C_8$ using $8$ letters. I am not having the formula now. I will post when I have it.
Note that for general $SO_{2^k}$ the dimension of $\gimel$ is $[2^{k-1}(2^k-1)-2^{k-1}]/2$=$2^{2k-2}-2^{k-1}$=$2(2^{k-2}(2^{k-1}-1))$=$2*dim(SO_{2^{k-1}})$. It is the number of roots, so it is interesting set to analyze from Lie groups perspective. I wonder how we could define such a set in any Lie group.
