To explore the formation of noncollinear magnetic configurations in materials with strongly correlated electrons, we derive a noncollinear
LSDA
+
U
model involving only one parameter
U
, as opposed to the difference between the Hubbard and Stoner parameters
U
−
J
. Computing
U
in the constrained random phase approximation, we investigate noncollinear magnetism of uranium dioxide
UO
2
and find that the spin-orbit coupling (SOC) stabilizes the
3
k
ordered magnetic ground state. The estimated SOC strength in
UO
2
is as large as 0.73 eV per uranium atom, making spin and orbital degrees of freedom virtually inseparable. Using a multipolar pseudospin Hamiltonian, we show how octupolar and dipole-dipole exchange coupling help establish the
3
k
magnetic ground state with canted ordering of uranium
f
orbitals. The cooperative Jahn-Teller effect does not appear to play a significant part in stabilizing the noncollinear
3
k
state, which has the lowest energy even in an undistorted lattice. The choice of parameter
U
in the
LSDA
+
U
model has a notable quantitative effect on the predicted properties of
UO
2
, in particular on the magnetic exchange interaction and, perhaps trivially, on the band gap: The value of
U
=
3.46
eV
computed fully ab initio delivers the band gap of 2.11 eV in good agreement with experiment, and a balanced account of other pertinent energy scales.