[SOLVED] Exercises on Lie-algebras I'm doing some homework on Lie-algebra's, and there are several things I am missing. Let me list them below in one thread. The statement is "If L is a classical linear Lie-algebra, then L is semisimple." I have reduced this to proving Rad L [itex]\subseteq[/itex] Z(L) -- then I have that Rad L = Z(L), so L is reductive and I can use an earlier result to conclude that Rad L = 0 so L is semisimple. But I don't know how to show this. Since what I need to prove does not hold in general, I should plug in some (common) property of classical linear Lie-algebras, but I can't really think of anything useful here... any ideas? Finally, I am required to prove that If any Lie-algebra L satisfies Rad L = Z(L), then all finite-dimensional representations of L in which Z(L) is represented by semisimple endomorphisms are completely reducible (Humphreys' book on Introduction to Lie-algebras and Representation theory, exercise 6.5d). So I took such a representation and found that I can simultaneously diagonalize all elements in the center, but I don't really see how this will give me any information on the reducibility of the entire representation. I know that I haven't really posted a lot of work, but I did try to reduce the problem to the points I'm really stuck at. I obviously don't expect a fully worked answer, but I'd really appreciate some hints or general steps to solve these problems. Thanks you very much!