  Using the Bismut's approach to Malliavin calculus, we introduce a simplified
Malliavin matrix ([11]) for stochastic differential equations (SDEs) force by
degenerate stable like noises. For the degenerate SDEs driven by Wiener noises,
one can derive a Norris type lemma and use it \emph{iteratively} to prove the
smoothness of density functions. Unfortunately, Norris type lemma is very hard
to be iteratively applied to SDEs with stable like noises. In this paper, we
derive a simple inequality as a replacement and use it to show that two
families of degenerate SDEs with stable like noises admit smooth density
functions. One family is the linear SDEs studied by Priola and Zabczyk ([13]),
under some additional assumption we can iteratively use the inequality to get
the smoothness of the density. The other family is the general SDEs with stable
like noises, we can apply this inequality only \emph{one} time and thus derive
that the SDEs admit smooth density if the \emph{first} order Lie brackets span
$\R^d$. The crucial step in this paper is estimating the smallest eigenvalue of
the simplified Malliavin matrix, which only uses some elementary facts of
Poisson processes and undergraduate level ordinary differential equations.
