I'll give it a stab but I'm just trying to help you decipher what I think
that decoction is doing...I'm not saying you should do a decoction this way but you'll get a ton of differing opinions on that.
First, you will have to adjust some things to make it right for your grain bill and your equipment. Decoctions take a little dialing in for each individual set-up. For example, using 22 qt. of 139.6 F water may or may not get you a rest temp of 134 F.
Also, when adding these hot liquids back to the mash do it in steps, stirring between each step, and monitor the temp to make sure you don't overshoot. If you reach your target temp and still have hot liquid left you'll have wait for it to cool (or cool it somehow) before adding the remainder.
1. Heat 22 Qt water to 139.6 F and add grains, stir, let rest 10 mins.
As mentioned you may have to adjust that strike water temp and you don't really need to rest for 10 minutes...once fully mixed you could start pulling the decoction immediately.
2. Take out 7.83 Qt of grain and boil for 30 min, then add back to mash to raise temp to 158║ for 90 min.
Don't worry about getting that exact volume, anywhere between 7.5-8 qt. should be close enough. You should convert the decoction first (just a quick, crude conversion...don't worry if it's not fully converted because you'll get it later) by raising it's temp to 158-160 F and resting there for 10 minutes or so...then boil it for 30 minutes. This decoction should be thick (mostly grain, I use a kitchen hand strainer) but not so thick you can't boil it without scorching, you may have to add some water during the boil to prevent it from getting too thick. IMO, 158 F is too hot and/or 90 minutes is too long. At that temp it'll convert in 10 minutes and make for a very dextrinous wort (which the decoction already does anyway). I would rest at a lower temp, prob no higher than 153 F and only for 60 minutes max.
3. Take out 4.71 Qt of grain, boil for 30 min, add back to mash to raise temp to 168║ for 10 min.
This decoction should be thin (mostly if not all liquid, I press a collander down into the mash and scoop out from inside that) and you don't really need to boil it 30 minutes...just to a good hot break is fine...10 minutes or so.
4. Sparge with 3.4 gal water at 170║
Standard but you might have to adjust your volume.