We consider the three-dimensional circular restricted three-body problem when the masless particle is orbiting one of the two primaries, the so called Lunar case. We compute a normal form up to orden eleven, that is, including in the expansion of the potential function polynomials of degree eleven when written in Cartesian coordinates.

We use Kustaaheimo-Stiefel regularisation technique
 transforming our perturbed Keplerian Hamiltonian into a polynomial perturbation of a harmonic oscillator in 1:1:1:1 resonance. Then, we apply Lie transformations, in particular Lunter's algorithm based on exponential maps, to compute the normal form of the Hamiltonian up to order eleven. Lunter's algorithm has the lowest computational cost of all the
norm form methods currently available.  This Lie transformation is equivalent to the elimination of the mean anomaly of the problem when expressed as a Keplerian system. After truncating higher-order terms we compute the invariants associated with the reduction process and write down the resulting Hamiltonian as a function of these invariants. This is performed through a polynomial reduction process that involves the computation of  Gr\"obner basis associated with the invariants and the constraints defined among them. Thus, the reduced Hamiltonian is defined in   the space generated by the invariants, which is $S^2\times S^2$.
 
We can perform a second Lie transformation which  is equivalent to the elimination of the argument of the node. We compute the associated invariants, a 
Gr\"obner basis related to them and to the existing contraints. Thus, we express the twice-reduced Hamiltonian function in terms of these invariants using Gr\"obner bases techniques. This Hamiltonian is defined on the twice reduced phase space which is homeomorphic with a sphere and that has two singular points when the third component of the angular momentum is zero.

The forward and backward changes of coordinates for the two transformations are done using Lunter's approach.

We detail the computational aspects needed to perform the overall process. All the computations can be made in a general purpose computer algebra system and we have used Mathematica. For instance the multiplication of polynomials is done using an efficient algorithm for sparse multivariate polynomials based on Kronecker substitutions. This allows to speed up the whole set of computations.

From a dynamical point of view, the approach can be used to analyse the motion of small particles (artificial or natural satellites) around planets or other objects, when the influence of a third body is strong enough to carry out many terms in the perturbations.