We explore several ways to dissect Brightest Cluster Galaxies (BCGs) and their surrounding Intracluster Light (ICL) using a surface brightness cut, a luminosity cut, excess light above a de Vaucouleurs profile, or a double Sersic decomposition. Assuming that all light above $M<-21.85~g~rm{mag}$ is attributable to the ICL, we find an average ICL fraction of $f^{rm MT}_{rm ICL}=71pm22%$ of all diffuse light centered on the BCG to belong to the ICL. Likewise, if we assume all light fainter than $rm{SB}>27$ $g$ mag arcsec$^{-2}$ to belong to the ICL, the average ICL fraction is $f^{rm SB27}_{rm ICL}=34pm19%$. After fitting a de Vaucouleurs profile to the inner parts of the SB profile, we detect excess light at large radii, corresponding to an average ICL fraction of $f^{rm DV}_{rm ICL}=48pm20%$. Finally, by decomposing the SB profile into two Sersic functions, we find an average ICL fraction of $f^{rm Stimes}_{rm ICL}=52pm21%$ associated with the outer Sersic component. Our measured ICL and BCG+ICL luminosities agree well with predictions from high-resolution simulations where the outer Sersic component traces the unrelaxed, accreted stellar material. BCG and ICL properties defined in this way are correlated with cluster parameters to study the co-evolution of BCGs, ICL, and their host clusters. We find positive correlations between BCG+ICL brightness and cluster mass, cluster velocity dispersion, cluster radius, and integrated satellite brightness, confirming that BCG/ICL growth is indeed coupled with cluster growth. On average, the ICL is better aligned than the BCG with the host cluster in terms of position angle, ellipticity, and centering. That makes it a potential Dark Matter tracer.
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