I'm trying to learn the mathematics behind spinors and I've come across sigma matrices and quaternions, both used to describe 3D rotations. Now it is possible to notice that any squared pair of sigma matrices returns the negative identity:
$$ \left( \sigma_i \sigma_j \right)^2 = - \mathrm{Id_2}, i \neq j$$
and any pair of sigma matrices anticommute with the "following" pair:
$$ \left( \sigma_i \sigma_j \right) \left( \sigma_j \sigma_k \right) = - \left( \sigma_j \sigma_k \right) \left( \sigma_i \sigma_j \right), i \neq j \neq k$$
Now these seem to be the same properties that characterize quaternion units $i$, $j$ and $k$:
$$ i^2 = j^2 = k^2 = -1 $$ $$ ij = -ji $$ $$ jk = -kj $$ $$ ki = -ik $$
Iin the source I'm using (which is this video), the author affirms that it is possible to associate sigma matrices and quaternion units in this way:
$$ i \to -\sigma_y \sigma_z $$ $$ j \to -\sigma_z \sigma_x $$ $$ k \to -\sigma_x \sigma_y $$
But isn't this association kind of arbitrary? For example this should also be a valid association:
$$ i \to -\sigma_x \sigma_y $$ $$ j \to -\sigma_y \sigma_z $$ $$ k \to -\sigma_z \sigma_x $$
Moreover, if this reasoning is correct, any association of this kind should be valid:
$$ i \to -\sigma_l \sigma_m $$ $$ j \to -\sigma_m \sigma_n $$ $$ k \to -\sigma_n \sigma_l $$
(with $l \neq m \neq n$). Now I get that the association proposed by the author of the video is the most intuitive one: a rotation on the $xy$ is kind of associated to a rotation about the $z$ axis, since the $k$ is associated to an expression where $\sigma_z$ doesn't appear. But, if my reasoning up until now is correct, is this the only justification for this association to be chosen among all the possible ones?
Thank you!