The Box Chain weave is  like a half byzantine. Once you have completed one side of the Byzantine weave, you just keep doing that over and over.

Here is a small sample done in gold with a splash of black. This is a great design for teenage men. I used 20 gauge 6mm rings.

Step 1. Pass 2 open rings (green) through 4 closed rings.

Step 2. Fold the last 2 rings back, green, split the center rings, and bring the folded rings together as shown.

Step 3. Add 2 more rings as shown above.

Step 4. Add 2 more rings, green.

Step 5. Fold those rings back, green.

Step 6. Add 2 more rings as shown and repeat steps 4-6.