  Galaxy clusters have a triaxial matter distribution. The weak-lensing signal,
an important part in cosmological studies, measures the projected mass of all
matter along the line-of-sight, and therefore changes with the orientation of
the cluster. Studies suggest that the shape of the brightest cluster galaxy
(BCG) in the centre of the cluster traces the underlying halo shape, enabling a
method to account for projection effects. We use 324 simulated clusters at four
redshifts between 0.1 and 0.6 from `The Three Hundred Project' to quantify
correlations between the orientation and shape of the BCG and the halo. We find
that haloes and their embedded BCGs are aligned, with an average $\sim$20
degree angle between their major axes. The bias in weak lensing cluster mass
estimates correlates with the orientation of both the halo and the BCG.
Mimicking observations, we compute the projected shape of the BCG, as a measure
of the BCG orientation, and find that it is most strongly correlated to the
weak-lensing mass for relaxed clusters. We also test a 2-dimensional cluster
relaxation proxy measured from BCG mass isocontours. The concentration of
stellar mass in the projected BCG core compared to the total stellar mass
provides an alternative proxy for the BCG orientation. We find that the
concentration does not correlate to the weak-lensing mass bias, but does
correlate with the true halo mass. These results indicate that the BCG shape
and orientation for large samples of relaxed clusters can provide information
to improve weak-lensing mass estimates.
