The quaternion algebra in the matrix representation is generated by the modified Pauli matrices iσ1, iσ2, iσ3, which in the representation of linear vector fields

iσ1=x4∂x1-x3∂x2+x2∂x3-x1∂x4

iσ2=x3∂x1+x4∂x2-x1∂x3-x2∂x4

iσ3=x2∂x1-x1∂x2-x4∂x3+x3∂x4

define pair rotations in 4-dimensional space. In turn, these linear vector fields are tangent vector fields to the Villarceau circles of classic 2-tori, lying on 3-spheres, because they specify the simultaneous rotation of the defining circles (lying in the orthogonal planes) of the torus and because they are orthogonal to the differential 1-form

x1dx1+x2dx2+x3dx3+x4dx4

However, if the paired rotations to replace the pair pseudo-rotations, which are a composition of pair rotations and equiaffine transformations of the classical 2-torus, the lie algebra of pseudo-rotations of the Villarceau circles will correspond to the lie algebra su(2).

Moreover, if we allow rotations and pseudo-rotations of the Villarceau circles, we get the lie algebras sl2( ℂ ).

Similarly, the algebra of rotations (pseudo-rotations) of the Willarceau circles of 4-tori lying on 7-spheres (7-hyperspheres of the spaces with a neutral metric) corresponds to the lie algebra sl4( ℂ ).