We note that the two results used essentially in this paper are:

(i) the one-to-one correspondence theorem of the extensions of a Riemannian foliation -having dense leaves and some type of sub-algebra of its Lie structural  being in [12],

(ii) “Existence of an equivalence between transversely diagonal foliation and transversely almost produced multi-foliated foliation of type  [10].

In this paper, we determine the nature of the Jacobian matrix of the transverse coordinate changer of every foliation of complete flag of extension

of a Riemannian foliation We prove that in case where  have dense leaves, each foliation of the flag  is transversely diagonal that is to say each foliation is defined by a foliated cocycle  such as on the open set there exists a coordinate system -transverse  such as the Jacobian  relatively  is diagonal.

transversally diagonal foliation, Riemannian foliation, extension of a foliation, complete flag of Riemannian extension, Riemannian having dense leaves, Lie structural algebra of a Riemannian foliation.