Yes I can.
In the end we built up the 6 beat kick by going through a 4 beat kick, although it wasn't a "real" 4 beat kick. The exercise we did was to immobilize one leg (with a fin) and to kick twice with the other leg on the catch of the hand on the same side and on the "push". There are three phases in the underwater stroke: the catch, the pull and the push - the idea is to synchronize a kick with each of these, the most important one being the catch (this is the Total Immersion kick which initiates the rotation of the hips). Then we moved to doing the same with the other leg and then with both legs (right right left left). The last stage was to insert an extra kick from the opposite leg in between these two:
1 .right arm catch: right leg kick
2. right arm pull: left leg kick
3. right arm push: right leg kick
4. left arm catch: left leg kick
5. left arm pull: left leg kick
6. left arm push: left leg kick
At first I was a bit wooden and my legs started to get tense and tired but, after relaxing into it, it became almost second nature. It reminded me a bit of learning to juggle where you start combining sequences of coordinated movements and then there is suddenly an "aha moment" when it all begins to flow of its own accord. Another thing altogether is whether I can do this in open water with hundreds of other people in race conditions!
Luis still reckons that my kick is much less propulsive than it can be. The next step is to try to kick from the hips.