The genus of the Coxeter–Todd lattice was described by (Scharlau & Venkov 1995) and has 10 isometry classes: all of them other than the Coxeter–Todd lattice have a root system of maximal rank 12.
Based on Nebe web page we can define K12 using following 6 vectors in 6-dimensional complex coordinates.
ω is complex number of order 3 i.e. ω3=1.
(1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), ½(1,ω,ω,1,0,0), ½(ω,1,ω,0,1,0), ½(ω,ω,1,0,0,1), By adding vectors having scalar product -½ and multiplying by ω we can obtain all lattice vectors.
Multiply it by 3 using multiplication by ω we obtain 756 unit vectors in K12 lattice.