Repeat the process of picking up a new bead, reversing direction, passing through the underlying bead, and then passing through the new bead again (Step 5 above), until you complete the second row.
Pick up the first bead for the third row (orange in the diagrams; Step 6 above) and then pass through the last bead in the second row and through the new bead again (Step 7 above). Pull the thread taut.
Pick up the second bead for the third row, pass through the bead below it in the opposite direction, and then pass through the new bead again (Step 8 above). Pull the thread taut.
Continue this process to stitch one bead at a time until you reach the end of the third row. Then reverse direction and stitch the first bead for the fourth row the same way you stitched the first bead in the third row.
Keep stitching one row at a time, back and forth, until you have your desired length of beadwork or until you complete your pattern.
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See an example of completed square stitch beadwork on page 3.