  In the past decade, there are many works on the finite element methods for
the fully nonlinear Hamilton--Jacobi--Bellman (HJB) equations with Cordes
condition. The linearised systems have large condition numbers, which depend
not only on the mesh size, but also on the parameters in the Cordes condition.
This paper is concerned with the design and analysis of auxiliary space
preconditioners for the linearised systems of $C^0$ finite element
discretization of HJB equations [Calcolo, 58, 2021]. Based on the stable
decomposition on the auxiliary spaces, we propose both the additive and
multiplicative preconditoners which converge uniformly in the sense that the
resulting condition number is independent of both the number of degrees of
freedom and the parameter $\lambda$ in Cordes condition. Numerical experiments
are carried out to illustrate the efficiency of the proposed preconditioners.
