  The bright central galaxies (BCGs) dominate the inner portion of the diffuse
cluster light, but it is still unclear where the intracluster light (ICL) takes
over. To investigate the BCG-ICL transition, we stack the images of
${\sim}3000$ clusters between $0.2{<}z{<}0.3$ in the SDSS $gri$ bands, and
measure their BCG+ICL stellar surface mass profile $\Sigma_{*}^{\texttt{B+I}}$
down to $3{\times}10^4\,M_{\odot}/\mathrm{kpc}^{2}$ at
$R{\simeq}1\,\mathrm{Mpc}$ (${\sim}32$ mag/arcsec$^2$ in the $r$-band). We
develop a physically-motivated method to decompose $\Sigma_{*}^{\texttt{B+I}}$
into three components, including an inner de Vaucouleurs' profile, an outer ICL
that follows the dark matter distribution measured from weak lensing, and an
intriguing transitional component between 70 and 200 kpc. To investigate the
origin of this transition, we split the clusters into two subsamples by their
BCG stellar mass $M_*^{\mathrm{BCG}}$ (mass enclosed roughly within 50 kpc)
while making sure they have the same distribution of satellite richness. The
$\Sigma_{*}^{\texttt{B+I}}$ profiles of the two subsamples differ by more than
a factor of two at $R{<}50\,\mathrm{kpc}$, consistent with their 0.34 dex
difference in $M_*^{\mathrm{BCG}}$, whereas on scales beyond 400 kpc the two
profiles converge to the same amplitudes, suggesting a satellite-stripping
origin of the outer ICL. Remarkably, however, the discrepancy between the two
$\Sigma_{*}^{\texttt{B+I}}$ profiles persist at above $50\%$ level on all
scales below 200 kpc, thereby revealing the BCG sphere of influence with radius
$R_{\mathrm{SOI}}{\simeq}$ 200 kpc. Finally, we speculate that the surprisingly
large sphere of influence of the BCG is tied to the elevated escape velocity
profile within $r_s$, the characteristic radius of the dark matter haloes.
