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Recently, lattice-reduction-aided detectors have been proposed for multiple-input multiple-output (MIMO) systems to give performance with full diversity like maximum likelihood receiver yet with complexity similar to linear receiver. However, these lattice-reduction-aided detectors are based on the traditional LLL reduction algorithm that was originally introduced for reducing real lattice basis, even though the channel matrices are inherently complex-valued. In this paper, we introduce the complex LLL algorithm for direct application to the channel matrix which naturally defines the basis of a complex lattice. Simulation results reveal that the new complex LLL algorithm can achieve a saving in complexity of nearly 50% over the traditional LLL algorithm, when applied to MIMO detection. We also show that the algorithm can further be accelerated by pre-ordering the basis vectors prior to the lattice reduction. It is noteworthy that the complex LLL algorithms aforementioned incur negligible bit-error-rate performance loss relative to the traditional LLL algorithm