Nope, I don't think you can do it in 2x2. Check the brioche stitch site
, it's similar and there's a few variations.
By knitting the WS row, or repeating the k1, k1b, you create a sort of garter stitch which makes the k1 sts appear to be purled on the RS every other row, so it looks similar to a k p rib, or a mock rib. (which is k1 p1 on the RS, knit all sts on the WS).