The half butterfly guard has been growing in popularity lately, possibly due to how well it joins the butterfly and the recently popularized x-guard, and how it can be used to counter the brabo choke, which has enjoyed wide-spread success against the traditional underhooking half guard game.
Here the butterfly hook is used to recover and sweep from being smashed in half guard.
All credit for this move goes to my instructor Eduardo de Lima of Gracie Barra Tampa (www.tampabjj.com). He's a great teacher and I owe everything to him.
I'm sorry I didn't get the other angle so you could see, but you'll need to take my word that he has the underhook on the side of his trapped leg, which prevents me from taking his back, and he is using his shoulder to flatten me out on my back. This is a bad place to be.
Note how I've got my legs closed: Inside leg over outside leg.
If I locked it the other way, he would be able to circle his trapped foot around mine and bring it up to my butt which can prevent me from doing the next step.
I open my legs enough to step both of my feet on his calf, with my toes pointing into each other.
I use "monkey feet" (as Eduardo calls this) to stretch out his leg, pulling his knee back.
I now circle my outside leg with the goal of hooking it under his thigh. That's why I pulled his knee back: I need to create space for my foot to come up and hook under his knee.
My inside foot stays across the back of his calf to keep his leg down.
If he keeps tight, or you couldn't create enough space, or you have freakishly long legs like me, you may need to escape your hips out to the side so you have a better angle to stick your hook in.
Sometimes even that is not enough, and you'll find you can only get the tips of your toes against his knee. If this happens, you need to do what Eduardo call the "snail move" with your toes, where you inch them along mat and wiggle them to get them dug under his leg.
If you had to escape your hip to get the hook, make sure you scoot back underneath him, since you'll need to be straight under him for the sweep.
Here you can see the hook from the other angle.
Your job is now to trap his underhooking arm so he cannot post with it when you sweep him. You can accomplish this a couple ways. Here I'm grabbing his belt and clamping his arm against my side with my elbow.
Alternate Grip 1
Or you can reach inside and grab his opposite lapel.
Alternate Grip 2
Or if it were no-gi, you could just hold his arm really tight.
I'm afraid I didn't get a picture that shows how I ended up here, but I think you can understand it: With his arm now trapped, I dove under his outside knee and pulled his leg up on to my shoulder.
Then (as you see here) I pulled him on top of me, using my hook some to control his weight.
This step and the next need to be done pretty quickly.
With him pulled on top of me, I take my inside foot off the back of his calf and post it on the ground to help as I...
...kick my hook into the air and turn into him.
Continuing with the motion, I've now swept him to his back.
Keep your hooking toes against his thigh. I'll explain why next.
Because I really wanted to stress this "toes on the leg" point, I got it from another angle. These toes are serving an important purpose -- they are keeping him from scrambling for his own half guard. You're going to keep your toes on his leg until you...
...bring your knee against his leg. Now it is blocking his leg, and you're free to take your toes off.
These may seem like a minor points, but it can be the difference between just reversing who's on the top of half guard or sweeping to side control.
Step 13 - Different Angle
Here's the previous step of blocking with the knee, just from the original angle.
Come to your knees. I've already posted my top leg since I like that side control.
Notice how my arm is still under his knee. We need to take that out of there quick so we don't end up in any dumb reverse triangle junk.
Here I've taken my arm out from between his legs and secured head-and-arm side control.
The finish from the other angle so you can see how I have my arms, head and chest.