  Given a point $A$ in the convex hull of a given adjoint orbit
$\mathcal{O}(F)$ of a compact Lie group $G$, we give a polynomial time
algorithm to compute the probability density supported on $\mathcal{O}(F)$
whose expectation is $A$ and that minimizes the Kullback-Leibler divergence to
the $G$-invariant measure on $\mathcal{O}(F)$. This significantly extends the
recent work of the authors (STOC 2020) who presented such a result for the
manifold of rank $k$-projections which is a specific adjoint orbit of the
unitary group $\mathrm{U}(n)$. Our result relies on the ellipsoid method-based
framework proposed in prior work; however, to apply it to the general setting
of compact Lie groups, we need tools from Lie theory. For instance, properties
of the adjoint representation are used to find the defining equalities of the
minimal affine space containing the convex hull of $\mathcal{O}(F)$, and to
establish a bound on the optimal dual solution. Also, the Harish-Chandra
integral formula is used to obtain an evaluation oracle for the dual objective
function. While the Harish-Chandra integral formula allows us to write certain
integrals over the adjoint orbit of a Lie group as a sum of a small number of
determinants, it is only defined for elements of a chosen Cartan subalgebra of
the Lie algebra $\mathfrak{g}$ of $G.$ We show how it can be applied to our
setting with the help of Kostant's convexity theorem. Further, the convex hull
of an adjoint orbit is a type of orbitope, and the orbitopes studied in this
paper are known to be spectrahedral. Thus our main result can be viewed as
extending the maximum entropy framework to a class of spectrahedra.
