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We derive a new multistage sphere decoding (MSD) algorithm, which is a generalization of the conventional sphere decoder (SD). This new MSD exploits that many higher order signal constellations can naturally be decomposed into several lower order constellations. We develop a two-stage SD for a 16-ary quadrature amplitude modulation (16QAM) multi-input multi-output (MIMO) system by decomposing 16QAM into two 4QAM constellations. The first stage generates a list of 4QAM vectors. For each of these, the second stage computes an optimal 4QAM vector. In the low signal-to-noise ratio (SNR) region, our MSD performs close to the original (single-stage) SD, but it has a lower complexity. In the high SNR region, our MSD is not suitable for reaching near maximum likelihood (ML) performance.