Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the big algebra and the root spaces have dim 1, summing to twice the dimension of the big algebra. This is because the size of the basis of the big algebra determines the number of roots which are the duel to the cartan sub algebra. Thus the cartan has the same dimension as the big algebra.
Help would be greatly appreciated.