By Irving Kaplansky

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Hence the n+1 unknowns v a n ish , and in p a rtic u la r Bn = 0. T h eorem 45 . Let L be a fin ite -d im e n s io n a l Lie a lg e b ra o v e r an a lg e b r a ic a lly c lo se d fie ld . Let u be a re g u la r elem en t of L. Let H be the su b alg eb ra c o rresp o n d in g to the c h a r a c te ris tic ro o t 0 of Ru . Then H is a C arta n su b a lg e b ra of L. P r o o f. Let L =L o +L a +L 0 +... be the d ecom p osition of L r e la tiv e to R^, and note that we a re w ritin g H fo r L^. In o r d e r to p ro ve H nilp oten t it su ffic e s , by E n gel's th e o re m (T h eo rem 13), to p ro ve th at a ll rig h t m u ltip lica tio n s by e le 44 m ents of H a re n ilp oten t when th ey act on H.

Potent. 2 S in ce R^ C R, R^ is Note th at bx € R C R^ and that by T h eorem 27, S^x is n il- B y T h eorem 29, applied to the im age of R^, we deduce that S S, is nilpotent and hence has tra c e a bx 0. We conclude the sectio n w ith an analogue of T h eorem 23. 28 T h eorem 3 1 . Let L be a fin ite -d im e n s io n a l L ie a lg e b ra o v e r a fie ld of c h a r a c te ris tic 0. Suppose that L ad m its a fin ite -d im e n s io n a l re p re se n ta tio n S fo r w hich the d e riv e d fo rm f is n o n -sin g u la r.

That H is a C arta n su b a lg e b ra . T h eorem 44. We p ro ve in T h eorem 45 T h eorem 4 4 is an e le m e n ta ry p re lu d e . Let A and B be n by n m a tr ic e s o v e r any fie ld . Suppose that kA + B is n ilp oten t fo r n + 1 d istin c t s c a la r s k. Then B is nilp oten t. P r o o f. On expanding (kA + B)n = 0 we get knA n + k11"1 (BAn_1 + A B A n "2 + . . + A n - 1 B) + . . + Bn = 0. This g ives us n+1 lin e a r hom ogeneous equations in n+1 "unknowns". The m a trix o f c o e ffic ie n ts is a V anderm onde m a trix and th e r e fo r e non sin g u la r.