  Let ${X_j},{Y_j}(j = 1, \cdot \cdot \cdot,n)$ be vector fields satisfying
H\"{o}rmander's condition and ${\Delta_L} = \sum\limits_{j = 1}^n {(X_j^2 +
Y_j^2)}$. In this paper, we establish some inequalities of Dirichlet
eigenvalues for degenerate elliptic partial differential operator ${\Delta_L}$
and $\Delta_L^2$. These inequalities extend Yang's inequalities for Dirichlet
eigenvalues of Laplacian to the settings here and the forms of inequalities are
more general than Yang's inequalities. To obtain them, we give a generalization
of the inequality by Chebyshev.
