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It is not always clear what one means by 'the simplest description' of one of the exceptional Lie groups. In the examples you've given above, you quote descriptions of these groups as automorphisms o …

There's an easy argument based on the Cayley transform: If $a$ is a skew-symmetric $n$-by-$n$ real matrix, then $I_n+a$ is invertible (since $(I_n-a)(I_n+a)=I_n-a^2$ is a positive definite symmetric …

Here is an algorithm to compute the Lie algebra of the group of isometries of a homogeneous space $G/H$ endowed with a $G$-invariant (pseudo-)Riemannian metric $g$. It is phrased in terms of essentia …

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 …

My guess is that Weinstein was thinking of this fact, but didn't get it out correctly: For every $n\not=15$, there is a compact Lie group $H_n\subseteq\mathrm{SO}(n{+}1)$ that acts transitively on the …

Yes. Here is one way: Consider standard $\mathbb{R}^3$ endowed with the Lorentzian quadratic form $Q = x^2+y^2-z^2$, and let $G\simeq\mathrm{O}(2,1)\subset\mathrm{GL}(3,\mathbb{R})$ be the symmetry …

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 …

Answers to the OP's question depend on where the OP is willing to start. To prove that an abstractly defined group is (i.e., has the structure of) a Lie group, one will have to use something nontrivi …

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 …

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 …

In the OP's particular case, the situation is somehwat simpler than the general case that José discusses. That's because the family of left-invariant metrics on $\mathrm{SU}(4)$ that the OP wants to …

Let me outline a different approach, namely the one É. Cartan himself took, one that doesn't depend on assuming completeness, doesn't rely on ideas about real-analyticity, and side-steps many of the i …

It seems that you are asking for descriptions of the groups $\mathrm{Spin}(p,q)$ as algebraic groups. This can certainly be done explicitly for low values of $p$ and $q$, but I don't know a general p …

Looking around on the internet, I found an English translation of Lie's 1891 paper Die Grundlagen für die Theorie der unendlichen kontinuierlichen Transformationsgruppen. I. (I.e., The foundations of …

I think you might want to read a couple of articles on the moving frame that carefully discuss this issue (and show that it is more subtle than most people realize). The first is a paper by Mark Green …