  $\mathfrak{L}_{II}$ operator is introduced by Y.-L. Xin (\emph{Calculus of
Variations and Partial Differential Equations. 2015,
\textbf{54}(2):1995-2016)}, which is an important extrinsic elliptic
differential operator of divergence type and has profound geometric meaning. In
this paper, we extend $\mathfrak{L}_{II}$ operator to more general elliptic
differential operator $\mathfrak{L}_{\nu}$, and investigate the clamped plate
problem of bi-$\mathfrak{L}_{\nu}$ operator, which is denoted by
$\mathfrak{L}_{\nu}^{2}$, on the complete Riemannian manifolds. A general
formula of eigenvalues for the $\mathfrak{L}_{\nu}^{2}$ operator is
established. Applying this formula, we estimate the eigenvalues with lower
order on the Riemannian manifolds. As some further applications, we establish
some eigenvalue inequalities for this operator on the translating solitons with
respect to the mean curvature flows, submanifolds of the Euclidean spaces, unit
spheres and projective spaces. In particular, for the case of translating
solitons, all of the eigenvalue inequalities are universal.
