  We extend the 3-point intrinsic alignment self-calibration technique to the
gravitational shear-intrinsic ellipticity-intrinsic ellipticity (GII)
bispectrum. The proposed technique will allow the measurement and removal of
the GII intrinsic alignment contamination from the cross-correlation weak
lensing signal. While significantly decreased from using cross-correlations
instead of auto-correlation in a single photo-z bin, the GII contamination
persists in adjacent photo-z bins and must be accounted for and removed from
the lensing signal. We relate the GII and galaxy density-intrinsic
ellipticity-intrinsic ellipticity (gII) bispectra through use of the galaxy
bias, and develop the estimator necessary to isolate the gII bispectrum from
observations. We find that the GII self-calibration technique performs at a
level comparable to that of the gravitational shear-gravitational
shear-intrinsic ellipticity correlation (GGI) self-calibration technique, with
measurement error introduced through the gII estimator generally negligible
when compared to minimum survey error. The accuracy of the relationship between
the GII and gII bispectra typically allows the GII self-calibration to reduce
the GII contamination by a factor of 10 or more for all adjacent photo-z bin
combinations at $\ell>300$. For larger scales, we find that the GII
contamination can be reduced by a factor of 3-5 or more. The GII
self-calibration technique is complementary to the existing GGI
self-calibration technique, which together will allow the total intrinsic
alignment cross-correlation signal in 3-point weak lensing to be measured and
removed.
