I used Paint Shop Pro (basically an economical PhotoShop).
1. Rotated the original square tile 45 degrees (had white background).
2. "Magic Wand" selected the now diamond shaped tile.
3. Pasted using Transparent Selection option - aligning to the upper-right edge (the rest of the copied tile "hangs off" the working area).
3. Pasted 3 more times using Transparent Selection option - aligning to the lower-right edge, lower-left and upper-left.
4. Used the Cropping tool to define the final "square".