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To amplify the points made by Laurent Berger, the literature I've seen (dating from around 1950) always specifies perfect fields; so I believe it was understood very early that Jordan-type decompositi …

"Classification" can mean more than one thing, but it's useful to be aware of the extensive development of adjoint quotients by Kostant, Steinberg, Springer, Slodowy, and others. This makes sense fo …

This is all very classical and dealt with in numerous books on Lie algebras. For instance, analyzing finite dimensional representations depends on complete reducibility and the (easy) explicit const …

If you look only at a simple Lie algebra, no two "adjacent" simple roots in the Dynkin diagram can form a right angle: being joined by at least one edge forces a different angle. In the simple case …

Ben has addressed the general question here, which as he suggests is not "standard" linear algebra. I guess this set-up might be relevant in categories of modules with fewer finiteness restrictions t …

The question is ambiguous in prime characteristic, since the meaning of "semisimple element" isn't straightforward for a simple Lie algebra consisting of matrices. Possibly what's meant here is "sem …

In terms of matrix theory, the question I'm led to is the following: Start with an $n$-dimensional vector space over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, which has …

Maybe it's helpful to add a longer comment, in community-wiki format. The original question is not well-formulated, I think, as shown in the later convoluted remarks on the case $\theta =1$. It's p …

To fill in the comments, there are basically two serious issues involved. 1) You want to know that every subgroup of finite index in $\mathrm{SL}(3,\mathbb{Z})$ contains some congruence kernel: the k …

In the wider setting of simple Lie algebras over $\mathbb{C}$, you are looking for the adjoint orbits of a Lie algebra of type B or D (odd or even orthogonal case). While this can be viewed concretely …

The answer to the question in the header is probably "no", but it's hard for me to interpret precisely what is being asked. Classically, the methods of Jacobson and Morozov, Kostant, Slodowy, and ot …

In an irreducible representation (finite or infinite dimensional), every nonzero vector is cyclic. This has nothing to do with Lie algebras as such, as it is true over an arbitrary ring. Thus for e …

A more "modern" book than those already mentioned is the one by Paul Halmos here. This was first published in 1942 in the Annals of Math. Studies series, with a later edition in 1958; that edition …

As I pointed out in my comment, there are too many questions listed here. Maybe I can clarify the term "Jordan-Chevalley decomposition" in the last one. Besides the arXiv post by Danielle Couty an …

As people have pointed out, the question raised in the Edit has a negative answer. To clarify this, start with $G = \mathrm{GL}_n(K)$ (for an algebraically closed field $K$) and embed $G$ (or any cl …