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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 $\mathbb{R}^7$. For simplicity, I'll do this for the split-form, …

Maybe the simplest argument, if you know something about compact Lie groups, is that SO(4) and SU(3) both have rank 2, i.e., they each contain a maximal torus, which is $S^1\times S^1$. Since all max …

The answer is 'no', the affine symplectic group cannot appear as a Lie subgroup of any compact Lie group. The reason is that the affine symplectic group contains $\mathrm{SL}(2,\mathbb{R})$ as a Lie …

You really should have a look at F. Reese Harvey's book Spinors and Calibrations, where all of your questions are answered. For example, your 'inclusion' (1) is not correct for sufficiently large $n$. …

You can work out the answers to these questions using the material in Chapter 11 of the book Spinors and Calibrations by F. Reese Harvey. You will also need to recall that, for $N\not=4$, the group o …

Yes, $G_1G_2\subset\mathrm{SU}(8)$ is an algebraic set. Here is the argument: Let $G_1{\times}G_2$ act on $\mathrm{SU}(8)\subset\mathrm{End}(\mathbb{C}^8)\simeq\mathbb{C}^{64}$ by the rule $(g_1,g_2) …

The Grassmannian $$\mathrm{Gr}(2,\mathbb{C}^n) = \frac{\mathrm{SU}(n)}{\mathrm{S}\bigl(\mathrm{U}(2){\times}\mathrm{U}(n{-}2)\bigr)}$$ is the compact Hermitian symmetric space of type AIII of rank $r …

This is only a partial answer, and it's based on YCor's comment about the groups that act transitively on spheres. What is missing, as YCor commented, is knowing that if $G\subset\mathrm{GL}(n,\mathb …

Your quote about Cartan thinking of $B_n$ and $D_n$ as 'projective groups..." is actually Cartan describing the lowest dimensional homogeneous space of these groups (except, of course, for a few excep …

I don't know that the partition has a name, so to speak, but it is well-understood and falls into the classification of the symmetric spaces of type A. Namely, those of type AI, which are $\mathrm{SU …

I believe that the answer to your question "Can $f$ be even and odd at the same time?" is no, but the argument that I have seems more complicated than I expected it to be. The idea is synthetically t …

When the (general) question is rephrased in less basis-dependent language, I believe that it translates to this: Let $\mathrm{U}(d)$ act on $V = \mathbb{C}^d$ in the usual way, and consider the $n$-f …

Here is a different characterization of the subgroup $\mathrm{Ad}\bigl(\mathrm{SU}(n)\bigr)\subset\mathrm{SO}(n^2{-}1)$ that works when $n>2$. Define a skew-symmetric trilinear form $\kappa:{\frak{ …

New answer: I now have an answer for the subgroup case that the OP originally asked about. In fact, one has the following result: Let $G$ be a connected compact Lie group and let $p = (p_1,\ldots,p …

Actually, it does not look like that. Take the case $N=3$. The representation of $\mathrm{SU}(3)$ on ${\frak{so}}(8)$ breaks up into the $8$-dimensional subspace ${\frak{su}}(3)$ and an irreducible …