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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, …

The $\sigma_i$ as defined above satisfy $\mathrm{d}\sigma_i = -2\,\sigma_j\wedge\sigma_k$ when $(i,j,k)$ is an even permutation of $(1,2,3)$. For later use, let $E_i$ be the dual (left-invariant) fram …

The group $H$ acts transitively and primitively on $\mathbb{C}=\mathbb{R}^2$. ('Primitive' means that $H$ preserves no nontrivial foliation.) It's a consequence of the classification of transitive pr …

Remark: I assume that you want $A$ to be a non-trivial bundle. Otherwise, of course, any parallelizable compact manifold would be an example. In particular, any compact Lie group would be an exampl …

This fails already for the split form of $\mathrm{G}_2$. Every finite-dimensional irreducible representation of its Lie algebra is a constituent of a tensor power of the $7$-dimensional representation …

I don't know a formula for $d(e^x,e^y)^2$, and I suspect that there is no simple formula, but the answer to Q2' is 'no'. The right hand side of (3) is linear in $Q^{-1}$, but it is not hard to see th …

Just so there'll be an answer: Whether every vector bundle over $G/\Gamma$ admits a flat connection depends on the group $G$ and the subgroup $\Gamma$. For example, if $G=\mathrm{SU}(2)\simeq S^3$ an …

This is something of an exercise is unwinding the definitions, but there's an interesting twist to it as well, so here's an outline of an answer: Let $G$ be a connected $3$-dimensional Lie group (not …

Here are a few brief comments, but, as you suspect, an enormous amount is known about the Laplacian on functions and forms on compact Lie groups. • Presumably, you know that the Killing form is non-de …

The result that you are looking for is not in Élie Cartan's 1936 book La topologie des groupes de Lie because it was not known to be true at the time the book was written. Indeed, as Cartan remarks i …

The answer is 'no'. In fact, a stronger statement is true: If $V$ is a finite dimensional vector space and $G\subset\mathrm{GL}(V)$ is a (connected) non-compact simple Lie group, then the only bounde …

This is more an answer to the corresponding question for the quaternions: Can one show that, for unit quaternions $p,q\in S^3\subset\mathbb{H}\simeq\mathbb{R}^4$, one has the inequality $$ \langle\lo …

Now that I have written out the completely elementary proof above for the quaternions and for $\mathrm{SO}(3)$, I feel that I should point out that the statement $|\log(ab)|\le |\log(a) + \log(b)|$, s …

I realized a problem with my 'counterexample', so I no longer claim that the desired inequality does not hold on $\mathrm{SO}(3)$. The proof that it does hold on the quaternions is still OK. I'll po …

There cannot be an 8-dimensional group $G$ acting effectively on $S^4$ by Riemannian isometries. The following argument may not be the best, but it explains why this is true. (I will assume that $G$ …