I'm reading chapter 11.2 of the Cheng and Li textbook 'Gauge theory of elementary particle physics'. It says that $T_+$, $T_-$ and $Q$ do not form a closed algebra. In order to fix this problem the theory is extended to contain extra fermions.
As Georgi and Glashow did, a heavy lepton $E$ is added to form a triplet together with a electron neutrino and electron. My question is about the commutator relation of $T_+$ with $T_-$. Can someone explain how they get to this commutator relation? In the book they say this relation is straighforward but I can't see how they get it. See the pictures for clarity.