The product of two nonscalar conjugacy classes in SL(n,K) contains matrices of arbitrary trace if n≥4 for any field or n=3 for finite fields.
Newman, Similarity over SL(n,F)
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On the traces of the product of 2 linear similarity classes
The product of two nonscalar conjugacy classes in SL(n,K) contains matrices of arbitrary trace if n≥4 for any field or n=3 for finite fields.