Elfweave is an interesting chain weave that bears visual similarities to the Box chain. However, its manner of construction is entirely different, as it is most easily made by creating a 2-1 chain and then modifying the chain into Elfweave. There is also a sheet version of the weave. Recommended aspect ratios are in the 3.5 range for the chain.

Step 1: Build a 2-1 chain a bit more than twice as long as you want the eventual finished piece to be. Find the middle link (in this image, colored copper), and pull it out to one end so the rest of the links are paired up. Try to line up the links in the chain as shown in the image; at the very least, you'll need the ones closest to the middle to be as shown.

Step 2: We're going to go down the chain and lock in adjacent pairs of rings. Take a new ring and pass it through two adjacent rings on one side of the chain, and then their counterparts on the other side. If you line up your chain so that the base (the first copper ring) is close to you, then you should start on the right-hand side. This way, when you're done adding the ring, the closure is in an easily-accessible location.

Step three: Now just keep adding rings like this down the length of the chain.

Step four: We're going to have to do the same thing to the underside of the chain, so turn the chain over. No rings are added in this step.

Step five: Just like in step two, add a new ring through the four rings closest to the end of the chain.

Step six: And just like in step three, continue adding rings down to the end of the chain.

Once you have a reasonably-sized chain made, you can add length to it relatively easily. Just find your original 2-1 chain and lengthen it, then connect the new sections with more copper-colored rings.
Finished Elfweave chain: