  Let $P$ be a Poisson algebra with a Lie bracket $\{, \}$ over a field $\F$ of
characteristic $p\geq 0$. In this paper, the Lie structure of $P$ is
investigated. In particular, if $P$ is solvable with respect to its Lie
bracket, then we prove that the Poisson ideal $\mathcal{J}$ of $P$ generated by
all elements $\{\{\{x_1, x_2\}, \{x_3, x_4\}\}, x_5\}$ with $x_1,\ldots ,x_5
\in P$ is associative nilpotent of index bounded by a function of the derived
length of $P$. We use this result to further prove that if $P$ is solvable and
$p\neq 2$, then the Poisson ideal $\{P,P\}P$ is nil.
