  We perform the complete stability study of the model of chromo-natural
inflation (Adshead and Wyman '12), where, due to its coupling to a SU(2)
vector, a pseudo-scalar inflaton chi slowly rolls on a steep potential. As a
typical example, one can consider an axion with a sub-Planckian decay constant
f. The phenomenology of the model was recently studied (Dimastrogiovanni,
Fasiello, and Tolley '12) in the m_g >> H limit, where m_g is the mass of the
fluctuations of the vector field, and H the Hubble rate. We show that the
inflationary solution is stable for m_g > 2 H, while it otherwise experiences a
strong instability due to scalar perturbations in the sub-horizon regime. The
tensor perturbations are instead standard, and the vector ones remain
perturbatively small. Depending on the parameters, this model can give a
gravity wave signal that can be detected in ongoing or forthcoming CMB
experiments. This detection can occur even if, during inflation, the inflaton
spans an interval of size Delta chi = O (f) which is some orders of magnitude
below the Planck scale, evading a well known bound that holds for a free
inflaton (Lyth '97).
