The basic loop in loop chain is the same as the foxtail chain (and in fact the two names are synonymous, just as birdcage and byzantine are the same weave). However, weaving loop in loop with metal requires an entirely different approach, as you cannot rely on the rings to stretch. Morever, metal rings are not generally already closed into unbroken circles for your convenience. You'll have to solder or fuse them shut first.

You're going to want to make this chain only with soft wires. I recommend copper for practicing and fine silver (not sterling) for high-quality pieces. If you're soldering wire and you don't like silver spots on your copper wire, you can use nickel-silver instead, but that metal is much tougher than copper or silver and will be harder to form into links.

You'll use rings with seemingly-absurd proportions for this weave. Try 20 gauge 7/16" ID for starting out. 24 gauge 5/16" ID should also work, if you have tools small enough to form it. Onward...

Step 1: Cut your links and close them. You should use either a jeweler's saw or a tool that gives a shear cut. Bolt-style cuts (the ones that look like ><) should not be used as they'll be harder to solder cleanly.

Step 2: Solder or fuse the ring closed. I'm not going to get into a primer on using a torch to melt metal here.

Step 3: Stretch the ring into a smooth oblong shape. There are two ways to do this. The first is to put the ring around the jaws of a pair of tweezernose (or better, roundnose) pliers. Open the plier's jaws to stretch the ring. Use another pair of pliers to squeeze the ring against the jaws of the first pair. The second technique basically substitutes two mandrels of equal size for the first pair of pliers. The mandrels should be small; on the order of 3/32" or 1/16" (if you're using 1/2" ID rings). I personally use the first technique, which is faster. If you're using soft enough wire, then you won't need to do the squeezing bit; just stretch the ring around the jaws of a pair of pliers.

Step 4: Bend the link around a mandrel until it takes on a 'U'-shape. Use a 1/8" mandrel or something else in that range. Make certain that the ends of the link line up evenly, as a little distortion in this part of the chain will screw up the rest of the piece.

Step 5: Make another oval link and pass it through the openings in the first link. This new link may be squeezed slightly in the middle by the other link; this is fine. Try to keep it from turning sideways, though.

Step 6: Bend the new link into a 'U'-shape like its partner. However, this time you don't need a mandrel to bend it around; just squeeze it together while holding onto the first link.

Step 7: Just continue adding links in this fashion until the chain is as long as you like. Have fun!