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The complexified octonions satisfy $N(xy) = N(x)N(y)$, and the norm is a nondegenerate quadratic form on $\mathbb{C}{\otimes}\mathbb{O}$, considered as a complex vector space. 1) No. …

Harvey and Lawson, Calibrated geometries (Acta Math. 1982) proves this (i.e., that the stabilizer of $\Omega_0$ is isomorphic to the nontrivial double cover of $\mathrm{SO}(7)$) using properties of the octonions … Second you can find a proof that doesn't rely on knowledge of the octonions but instead relies on the classification of Lie algebras and groups in Bryant's Metrics with exceptional holonomy (Ann. of Math …

It might be immodest of me to mention my own work, but there is quite a lot known about the pseudo-holomorphic curves in $S^6$. For example, it is known that the pseudoholomorphic rational curves are …

-x\,I_8 }\ :\ \matrix{ x\in\mathbb{R},\cr {\bf w},{\bf x}\in\mathbb{O},\cr \ a\in{\frak{spin}}(8)}\ \right\} \subset{\frak{sl}}(16,\mathbb{R})\,, $$ where $C$ denotes conjugations in the octonions … , $R_{\bf w}$ denotes right multiplication by ${\bf w}$ in the octonions, $L_{\bf x}$ denotes left multiplication by ${\bf x}$ in the octonions, and the assignments $a\mapsto a_k$ for $k=1,3$ are Lie algebra …

Hence the automorphism group of the octonions is a Lie group of dimension $14$. Note that, any automorphism of the octonions that fixes three such elements is the identity. … imaginary octonions. …

All your questions (and much more) are answered in the 1982 paper Calibrated Geometries by Harvey and Lawson. See Acta Mathematica July 1982, Volume 148, Issue 1, pp. 47-157. Just so you'll know: H …

Let $a,b,c\in\mathbb{O}$ be octonions and consider the linear map $L:\mathbb{O}\to\mathbb{O}$ defined by $$ L(x) = (b(cx))a = R_aL_bL_c(x). $$ One desires a formula for the characteristic polynomial of …

I think that you want to look at the relevant passages in Spin Geometry by Lawson and Michelsohn, particularly Chapter IV, Sections 9 and 10, where they explain in general how the square of a spinor c …

Yes, in fact, any $2$-by-$2$ octonionic Hermitian matrix is equivalent under the natural $\mathrm{Spin}(9)$ action to a diagonal $2$-by-$2$ octonionic Hermitian matrix. This follows from the well-know …

Here's a description that doesn't use octonions; instead, it uses the definition of $\mathrm{G}_2$ as the stabilizer of a $3$-form on $\mathbb{R}^7$. …