Lattice-reduction-aided detection (LRAD) has been shown to considerably increase the performance of linear Multiple-Input Multiple-Output (MIMO) detection systems. In this paper, we show that conventional LRAD is inherently incompatible with the use of different modulation formats across transmit antennas or, more generally, the use of transmit power scaling. This motivates the first of two enhancements proposed in this paper, wherein we show how the MIMO channel model can be extended to cater for transmit power scaling. The second enhancement exploits temporal channel correlation to yield lattice reduction-based detectors exhibiting considerably reduced complexity. For typical mobile environments, for example, we demonstrate a complexity reduction approaching 33% on a 4 x 4 system, without compromising performance.