  In this paper we prove the generalized Kaplansky conjecture for the Jordan
algebras of the type $J_n$ in particular for self adjoint $2\times 2$ matrices
over $\R$, over $\C$, $\HH$ and $\Oct$. In fact, we prove that the image of
multilinear polynomial must be either $\{0\}$, $\mathbb{R}$, the space of pure
elements $V$, or $J_n$.
