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This is not true. Take $\mathrm{U}(2)$, for example. It is not the direct product of its center (the matrices of the form $\mathrm{e}^{i\theta} \mathrm{I}_2$) with the simple part $\mathrm{SU}(2)$. …

Just computing the exponential gives $$ \exp\left( \begin{matrix} 0 & 0 & 0 \cr x & 0 & -t\cr y & t & 0 \end{matrix} \right) = \left( \begin{matrix} 1 & 0 & 0 \cr x\frac{\sin t}{t}-y\frac{1{-}\cos t}{ …

One way to make the subgroup $\mathrm{Sp}(n)\cdot\mathrm{Sp}(1)\subset\mathrm{SO}(4n)$ explicit is to think of $\mathbb{R}^{4n}$ as $\mathbb{H}^n$, i.e. as columns of quaternions of height $n$, where …

As an answer to your weaker version: No. For example, $\mathrm{G}_2$ is a closed proper subgroup of $\mathrm{SO}(7)$, but it is not isomorphic to any subgroup of $\mathrm{SO}(6)$. Of course, this f …

No. A simple example is to let $G = \mathrm{SO}(6)$, which has rank $3$. Let $v_1$ and $v_2$ be unit orthogonal vectors in $\mathbb{R}^6$ and let $H_1\simeq \mathrm{SO}(5)\subset G$ be the stabilize …

Sure. Let $G$ be the group of diffeomorphisms of the real line. Then $\frak{g}$ is the Lie algebra of vector fields on the real line. However, many vector fields on the real line cannot be exponent …

Perhaps I'm misunderstanding your question, but what about the following example? Let $G = \mathrm{SO}(3)$ and let $K=\{e\}$ be the identity subgroup. Then $G/K = \mathrm{SO}(3)$ and $G_v = K$ for a …

This is not really an answer as much as it is a caution that the problem, even with the extra assumptions I was able to solicit from the OP, is not going to have a very nice answer unless one adds som …

Unfortunately, the answer is 'no'. You are basically asking whether you can simultaneously diagonalize two quadratic forms in three variables, and the answer is that, 'generically' you can (and you a …

One way to do this is to reduce this to understanding sets of points on the $2$-sphere up to a certain equivalence. If you let $\mathcal{H}_k$ denote the homogeneous polynomials $p$ of degree $k$ on …

But, can't you read this off the table of simple roots for each of the exceptional Lie algebras? For example, $\mathrm{G}_2$ contains $\mathrm{SU}(3)$ as a subgroup and the maximal torus of $\mathrm{ …

I guess that the answer to your first question is no, based on the following: If the union of the $G_i$ were dense in $G=\mathrm{Diff}(M)$, then, presumably, for $i$ sufficiently large, the action of …

NB: I have edited my answer to remove the comment at the beginning about correcting the original sign of the OP in the formula for $H$, since the OP has now corrected the erroneous sign in the questio …

The OP specifically asked for a(n ordinary) metric, not a Riemannian metric. While Misha and Paul have given good answers, I think that it's worth pointing out that, if one just takes an arbitrary le …

There won't be any `right' answer here because there are many different ways that one can come across groups in studying various algebra problems, but here are a few examples: Maybe the most famous e …