On a real analytic $5$-dimensional CR-generic submanifold
$M^5 \subset \mathbb{C}^4$ of codimension $3$ hence of CR dimension $1$,
which enjoys the generically satisfied nondegeneracy condition
\begin{align*}
{\bf 5}
&= \text{rank}_\mathbb{C} \big(
T^{1,0}M+T^{0,1}M +
\big[T^{1,0}M,\,T^{0,1}M\big] \,+
\\&\qquad
+ \big[T^{1,0}M,\,[T^{1,0}M,T^{0,1}M]\big]
+ \big[T^{0,1}M,\,[T^{1,0}M,T^{0,1}M]\big] \big),
\end{align*}
a canonical Cartan connection is constructed after reduction
to a certain partially explicit $\{ e\}$-structure
of the concerned local biholomorphic equivalence problem.
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