The stable mod 2 cohomologies of the spectra for connective real and complex K-theories are well known and easy to work with. However, the known bases are in terms of the anti-automorphism of Milnor basis elements. We offer simple bases in terms of admissible sequences of Steenrod operations that come from the Adem relations. In particular, a basis for H*(bu) is given by those Sq^I with I admissible and no Sq¹ or Sq²ⁿ⁺¹ appearing for n > 0.

We are grateful for the referee's very careful reading and many useful suggestions for improved exposition.