The Rolling Loop Choke
The fundamental loop choke is a sneaky, fast attack from open guard - both top and bottom! Making this move even more sneaky, several rolling variations have cropped up over the past 20 years or so, sometimes entering the world stage in dramatic fashion. Here are a few different versions you can try out, determining one or two that work best for you (or, just use all of them in combination).
From the Top
Rolling from the top can be a bit of a surprise for your partner, who is used to the somewhat more conventional idea that submissions happen from the guard when you're on the bottom, not from the top when you're working to pass. Start standing above your partner, with your right leg forward (similar to the "catching the trail arm" position). As your partner sits up, grip their cross lapel with your left hand, and give them a little push. As they react back toward you, "loop" your left elbow around their head, aiming to put your biceps on their back. Lean over them, putting some weight onto their back, so the circle is sealed. Next, grip their pants with your right hand, and drive your head through the gap that is created between their arm and hip area. Now just roll forward, taking care to tuck your chin and stay glued to your partner during the roll. Once you end up perpendicular with your partner, both with your hips facing up, you can finish the choke by doing a simple shrug motion while pulling their leg toward you. This is not unlike a backwards, upside down bow and arrow choke.
Same Move, From the Guard
When attacking this variation from the bottom, the main differences are in the beginning of the move. From a butterfly guard position, grab their collar once again with your left hand, loose enough so that you can loop your arm around their head. Sit up, and push their head with your free right hand, making sure it ends up below your left armpit. You'll need to come forward to your knees in order to complete this task. The gripping, rolling, and finishing is essentially identical from here.
Arm Trap Variation
Just like with basic gi chokes from the back, there are numerous analogous finishes from the loop choke position. Enter the same way, sitting up from the bottom and capturing their head. This time, as you roll through, either you lose the grip on their pants entirely, or else you find that you are unable to finish your partner using the previous techniques. Instead, use your right arm to thread in front of their arm, but behind their head. This "single wing" variation can become insanely tight, very quickly, so use judiciously.
As a Pass
Like the Kimura grip, the loop choke can be used to pass your partner's guard. Start with a basic rolling loop choke (if such a thing can be called "basic" in any context), and as your partner begins to turn away, use the preceding arm weave variation ("single wing") to try for the submission. This time, you find that your grip around their lapel isn't ideal for some reason, or you've lost it entirely. Use the arm thread concept to keep the inside arm isolated, helping to facilitate a solid guard pass technique.
Arm Triangle Version
An even more insidious version of this choke arises in much the same manner as the previous techniques. You are coming up on top, getting ready to roll through, but this time, instead of gripping their pants or going for the arm weave variation, reach across for their far arm. This anchor operates very much like an arm triangle (or "head and arm" choke), with their own shoulder and arm choking them. This version is shockingly tight.
Which loop choke do you prefer?
As you experiment with these variations, prepare to be loathed by your training partners for a while. If you get really good at these variations, you'll be able to set up the loop choke from a wide variety of positions, and you'll also be able to finish in spite of several different types of resistance. Keep in mind the fundamentals of any choke - cutting off the carotid arteries bringing blood to the brain; these all do an amazing job of "pushing the button" that makes people go to sleep. As always, let me know how these variations work out for you!