Multiple-input multiple-output (MIMO) systems are playing an important role in the recent wireless communication. The complexity of the different systems models challenge different researches to get a good complexity to performance balance. Lattices Reduction Techniques and Lenstra-Lenstra-Lovasz (LLL) algorithm bring more resources to investigate and can contribute to the complexity reduction purposes. In this paper, we are looking to modify the LLL algorithm to reduce the computation operations by exploiting the structure of the upper triangular matrix without big performance degradation. Basically, the first columns of the upper triangular matrix contain many zeroes, so the algorithm will perform several operations with very limited income. We are presenting a performance and complexity study and our proposal show that we can gain in term of complexity while the performance results remains almost the same.