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 r1/4, r1, and double r1/4 models. We obtain statistically superior fits using a two-component model consisting of a pair of r1/4 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 those of 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 μ-re relation than that of the elliptical fundamental plane. We interpret this outer component as a population of intracluster stars that trace the cluster potential. The two components are strongly aligned (|Δθ| 10°) 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.
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