▁theory ▁and ▁to ▁prov ing ▁the ▁existence ▁of ▁the ▁exception al ▁Lie ▁al gebras ▁belonging ▁to ▁each ▁of ▁the ▁types ▁of ▁simple ▁complex ▁Lie ▁al gebras ▁that ▁K illing ▁had ▁shown ▁to ▁be ▁possible . ▁Later ▁Cart an ▁completed ▁the ▁local ▁theory ▁by ▁explicitly ▁solving ▁two ▁fundamental ▁problems , ▁for ▁which ▁he ▁had ▁to ▁develop ▁entirely ▁new ▁methods : ▁the ▁classification ▁of ▁simple ▁real ▁Lie ▁al gebras ▁and ▁the ▁determ ination ▁of ▁all ▁ir reducible ▁linear ▁representations ▁of ▁simple ▁Lie ▁al gebras , ▁by ▁means ▁of ▁the ▁notion ▁of ▁weight ▁of ▁a ▁representation , ▁which ▁he ▁introduced ▁for ▁that ▁purpose . ▁It ▁was ▁in ▁the ▁process ▁of ▁determ ining ▁the ▁linear ▁representations ▁of ▁the ▁orthogonal ▁groups ▁that ▁Cart an ▁discovered ▁in ▁ 1 9 1 3 ▁the ▁spin ors , ▁which ▁later ▁played ▁such ▁an ▁important ▁role ▁in ▁quantum ▁mechan ics . ▁ ▁After ▁ 1 9 2 5 ▁Cart an ▁grew ▁more ▁and ▁more ▁interested ▁in ▁topological ▁questions . ▁Sp ur red ▁by ▁W ey l ' s ▁brilliant ▁results ▁on ▁compact ▁groups , ▁he ▁developed ▁new ▁methods ▁for ▁the ▁study ▁of ▁global ▁properties ▁of ▁Lie ▁groups ; ▁in ▁particular ▁he ▁showed ▁that ▁top olog ically ▁a ▁connected ▁Lie ▁group ▁is ▁a ▁product ▁of ▁a ▁E uclidean ▁space ▁and ▁a ▁compact ▁group , ▁and ▁for ▁compact ▁Lie ▁groups ▁he ▁discovered ▁that ▁the ▁possible ▁fundamental ▁groups ▁of ▁the ▁underlying ▁manifold ▁can ▁be ▁read ▁from ▁the ▁structure ▁of ▁the ▁Lie ▁algebra ▁of ▁the ▁group . ▁Finally , ▁he ▁out lined ▁a ▁method ▁of ▁determ