  We present a detailed analysis of the surface brightness distribution of the
brightest cluster galaxy (BCG) in each of 24 galaxy clusters at 0.03<z<0.13. We
use two-dimensional profile fitting to model the surface brightness out to
r=300 kpc for each BCG, comparing r^{1/4}, r^{1/n}, and double r^{1/4} models.
We obtain statistically superior fits using a two component model consisting of
a pair of r^{1/4} profiles with independent scale lengths, ellipticities, and
orientations. The two component model can simply reproduce the observed
position angle and ellipticity gradients, which cannot generally be explained
purely by triaxiality. The inner component of our two component model has
properties similar to a typical massive elliptical galaxy and is clearly
associated with the BCG. The outer component is 10-40 times larger in scale,
has ~10 times the total luminosity of the inner component, and exhibits a
steeper <\mu>-r_e relation than that of the elliptical fundamental plane. We
interpret this outer component as a population of intracluster stars tracing
the cluster potential. The two components are strongly aligned
(|\Delta\theta|<10\degr) in roughly 40% of the clusters. When they are not
aligned, the components tend toward high degrees of misalignment, suggesting
that accretion of infalling material may change the orientation of some BCGs
for a time. The extent of the outer component and its similar elongation to
published cluster galaxy distributions indicates that the evolution of the
intracluster light is tied to the cluster as a whole rather than to the BCG.
