In our previous paper, for a generic rank-2 vector distributions
on an n-dimensional manifold (n ≥ 5) we constructed a special
differential invariant, the fundamental form. In the case n = 5,
this differential invariant has the same algebraic nature, as the covariant
binary biquadratic form, constructed by E. Cartan, using his
“reduction-prolongation” procedure (we call this form the Cartan tensor).
In the present paper, we prove that our fundamental form coincides
(up to the constant factor −35) with the Cartan tensor. This
result explains geometrically the reason for the existence of the Cartan
tensor (originally, this tensor was obtained by very sophisticated algebraic
manipulations) and gives the true analogs of this tensor in the
Riemannian geometry. In addition, as a part of the proof, we obtain
a new useful formula for the Cartan tensor in terms of the structural
functions of any frame naturally adapted to the distribution.