why not? Does the choice matter at all? If we wanted to be more formal here we should write, I think, $d(dq):T(TI)\to T(TQ)$, where $dq(t,\lambda e_1)=(q(t),\lambda \dot q(t))$, and thus $$(\,\,d(dq)\,\,)((t,\lambda e_1), \,\, (\mu e_1, \eta e_1)\,\, ) = (dq(t,\lambda e_1), \,\, (\mu \dot q(t), \,\,\lambda\mu \ddot q(t)+\eta \dot q(t)) \,\,).$$ Ok this won't make any sense to anyone, let me try to explain wth I mean with it