A new 3M-dimensional Lie algebra X is constructed firstly. Then, the corresponding loop algebra X̃ is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra Ã₁. It follows that a general scheme for generating multi-component integrable hierarchy is proposed. By taking advantage of X̃, a new isospectral problem is established, and then well-known multi-component TC hierarchy is obtained. Finally, an expanding loop algebra F̃_M of the loop algebra X̃ is presented. Based on the F̃_M, the multi-component integrable coupling system of the generalized multi-component TC hierarchy has been worked out. The method in this paper can be applied to other nonlinear evolution equations hierarchies. It is easy to find that we can construct any finite-dimensional Lie algebra by this approach.