  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
S\'ersic decomposition. Assuming that all light above $M<-21.85~g'~\rm{mag}$ is
attributable to the ICL, we find an average ICL fraction of $f^{\rm MT}_{\rm
ICL}=71\pm22\%$ 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 $f^{\rm
SB27}_{\rm ICL}=34\pm19\%$. 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 $f^{\rm DV}_{\rm ICL}=48\pm20\%$. Finally, by
decomposing the SB profile into two S\'ersic functions, we find an average ICL
fraction of $f^{\rm S\times}_{\rm ICL}=52\pm21\%$ associated with the outer
S\'ersic component. Our measured ICL and BCG+ICL luminosities agree well with
predictions from high-resolution simulations where the outer S\'ersic 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.
