  We investigate the constraints that the LHC can set on a 126 GeV Higgs boson
that is an admixture of CP eigenstates. Traditional analyses rely on Higgs
couplings to massive vector bosons, which are suppressed for CP-odd couplings,
so that these analyses have limited sensitivity. Instead we focus on Higgs
production in gluon fusion, which occurs at the same order in the strong
coupling for both CP-even and -odd couplings. We study the Higgs plus two jet
final state followed by Higgs decay into a pair of tau leptons. We show that
using the 8 TeV dataset it is possible to rule out the pure CP-odd hypothesis
in this channel alone at nearly 95\% C.L, assuming that the Higgs is CP-even.
We also provide projected limits for the 14 TeV LHC run.
