  We use the spectroscopy and homogeneous photometry of 97 Type Ia supernovae
obtained by the \emph{Carnegie Supernova Project} as well as a subset of 36
Type Ia supernovae presented by Zheng et al. (2018) to examine maximum-light
correlations in a four-dimensional (4-D) parameter space: $B$-band absolute
magnitude, $M_B$, \ion{Si}{2}~$\lambda6355$ velocity, \vsi, and \ion{Si}{2}
pseudo-equivalent widths pEW(\ion{Si}{2}~$\lambda6355$) and
pEW(\ion{Si}{2}~$\lambda5972$). It is shown using Gaussian mixture models
(GMMs) that the original four groups in the Branch diagram are well-defined and
robust in this parameterization. We find three continuous groups that describe
the behavior of our sample in [$M_B$, \vsi] space. Extending the GMM into the
full 4-D space yields a grouping system that only slightly alters group
definitions in the [$M_B$, \vsi] projection, showing that most of the
clustering information in [$M_B$, \vsi] is already contained in the 2-D GMM
groupings. However, the full 4-D space does divide group membership for faster
objects between core-normal and broad-line objects in the Branch diagram. A
significant correlation between $M_B$ and pEW(\ion{Si}{2}~$\lambda5972$) is
found, which implies that Branch group membership can be well-constrained by
spectroscopic quantities alone. In general, we find that higher-dimensional
GMMs reduce the uncertainty of group membership for objects between the
originally defined Branch groups. We also find that the broad-line Branch group
becomes nearly distinct with the inclusion of \vsi, indicating that this
subclass of SNe Ia may be somehow different from the other groups.
