Operations in G3 have a bunch of extra work compared to operations on quaternions alone due to the presence of odd grades. However, if your values are quaternionic, these grades should always be zero, so you can safely elide the operations. The compiler just has to be smart enough to see this.
The basis of G3 is 1,x,y,z,xy,xz,yz,xyz. Quaternions only have 1,xy,xz,yz. Quaternion multiplications are closed; xyyz = xz, xyxz = -yz, xzyz = -xy. xyxy=-1. If your compiler is smart enough to figure out that x,y,z,xyz are always zero, you can save the extraneous operations.
The basis of G3 is 1,x,y,z,xy,xz,yz,xyz. Quaternions only have 1,xy,xz,yz. Quaternion multiplications are closed; xyyz = xz, xyxz = -yz, xzyz = -xy. xyxy=-1. If your compiler is smart enough to figure out that x,y,z,xyz are always zero, you can save the extraneous operations.