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Among the finite dimensional Lie algebras over a field of characteristic 0, there is a sensible definition of "reductive Lie algebra" going back at least to the 1960 first chapter of N. Bourbaki's tre …

The short answer is no. In prime characteristic, the Killing form sometimes behaves badly even for simple Lie algebras. If "semisimple" means that the solvable radical is zero, there is no way to …

Sasha has answered concisely the basic question here with a counterexample involving Lie type $A$, where all primes are good but need not be very good (meaning that $p$ should not divide $n$ for $\mat …

[EDIT] This replaces my less focused (and more optimistic) attempt at answering both questions. One remark about terminology: it's more standard to refer to the "smoothness" of a conjugacy class tha …

As my comment indicated, there is currently little hope of writing down general branching rules in characteristic $p$. In fact, given the history of work on Lusztig's conjecture about formal charact …

Roger Richardson amplified Kostant's result in characteristic 0, which in turn led Steve Donkin to work out a closely parallel version in prime characteristic: On conjugating representations and adjo …

The answer is fairly classical (and no longer really at research level), but the most natural setting for it is the more general study of Lie algebras obtained by reduction mod $p$ from Chevalley's in …

Probably the recent paper by Herpel and Stewart here helps to settle your basic question positively. Though the correspondence between subgroups of Lie groups and Lie algebras in the classical situa …