  A result of A. Joseph says that any nilpotent or semisimple element $z$ in
the Weyl algebra $A_1$ over some algebracally closed field $K$ of characterstic
0 has a normal form up to the action of the automorphism group of $A_1$. It is
shown in this note that the normal form corresponds to some unique pair of
integers $(k,n)$ with $k\ge n\ge 0$, and will be called the Joseph norm form of
$z$. Similar results for the symplectic Poisson algebra $S_1$ are obtained. The
Dixmier conjecture can be reformulated as follows: For any nilpotent element
$z\in A_1$ whose Joseph norm corresponds to $(k,n)$ with $k>n\ge 1$, there
exists no $w\in A_1$ with $ [z,w]=1$. It is known to hold true if $k$ and $n$
are coprime. In this note we show that the assertion also holds if $k$ or $n$
is prime. Analogous results for the Jacobian conjecture for $K[X,Y]$ are
obtained.
