The eigenvalues of the weakly coupled T ⊗ t Jahn-Teller problem are known for several decades, and the same holds also true for the eigenstates. These, however, are only given in the traditional position representation, which proves inconvenient if one attempts to extend the weak-coupling treatment into the region of stronger coupling. Here the solution of the T ⊗ t eigenvalue problem at weak coupling is derived in terms of creation and annihilation operators. This reformulation of the problem is nontrivial, since the algebraic form of the oscillator eigenvectors, being simultaneous angular-momentum eigenstates, has been worked out only recently and is probably still widely unknown. The electronic and oscillator eigenstates are then coupled to form eigenvectors of the total angular momentum. Finally, in preparation for an extension of the weak-coupling treatment, the action of the boson creation and annihilation operators on the oscillator eigenvectors is calculated, thus completing the algebraic approach to the weakly coupled T ⊗ t system.