M Theory Lesson 231

Let us relabel the $M_{i}$ as $X$, $Y$, $Z$ and $T$. Then the analogue of the quandle relations for the binary Lie bracket in the Pauli case are the cyclic ternary relations

$X = ZYTZ$
$Y = TZXT$
$Z = XTYX$
$T = YXZY$

which are easily verified by multiplying out the matrices. Note also how the two dual pairs yield the quantum plane relations

$YX = \omega XY = \omega (312)$
$TZ = \omega ZT = \omega (231)$

There are also other relations of this type. So the Lie bracket is not the natural structure to study for three dimensional quantum information.