Geodesic
Computation of the commutator of $2\times2$ matrices via six multiplications
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 9

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An algorithm for computing the commutator $AB-BA$ of $2\times2$ matrices $A$ and $B$ is proposed. The algorithm involves six multiplications. In the case of Hermitian $2\times2$ matrices, it involves six real multiplications. 
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     author = {A. Ya. Belyankov},
     title = {Computation of the commutator of $2\times2$ matrices via six multiplications},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1459},
     publisher = {mathdoc},
     volume = {47},
     number = {9},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_9_a0/}
}
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A. Ya. Belyankov. Computation of the commutator of $2\times2$ matrices via six multiplications. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 9. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_9_a0/