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The heuristic is as follows: The toral (and abelian) subgroup allows us to simultaneously diagonalize them, such that $C$ are all upper triangular matrices with ones on the diagonal (unipotent). Then $C_\alpha T=TC_\alpha$ is a Borel subgroup (maximal solvable). The fact that $\alpha$ is a positive root, fixes the root system and so the possible conjugates. ...


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