Let $f$ be an automorphism of the octonions algebra. Then $f(x)=x$ for $x\in \mathbb R$ and $f$ restricted to $Im \mathbb O$ is in $SO(7)$. By the properties of the rotations there is an orthonormal basis: $e_1,e_2,...,e_7$ in $\mathbb O$ and real numbers $\phi_1,\phi_2,\phi_3$ such that: $$ f(e_1)=e_1,\\ f(e_2)=\cos \phi_1 e_2-\sin \phi_1 e_3, \\ f(e_3)=\sin \phi_1 e_2+\cos \phi_1 e_3,\\ f(e_4)=\cos \phi_2 e_4-\sin \phi_2 e_5, \\ f(e_5)=\sin \phi_2 e_4+\cos \phi_2 e_5,\\ f(e_6)=\cos \phi_3 e_6-\sin \phi_3 e_7, \\ f(e_7)=\sin \phi_3 e_6+\cos \phi_3 e_7. $$
Does there exist a Cayley's triple $(i,j,l)$ in $Im \mathbb O$ (i.e. unit elements of $Im \mathbb O$ satisfying $l\bot i,j,ij$) such that $$ e_1=i,\\ e_2=j,\\ e_3=k,\\ e_4=l,\\ e_5=il,\\ e_6=jl,\\ e_7=(ij)l. $$