Let $M$ be a matrix in $M_{2}(\mathbf{R})$. Prove that $M$ is semi-simple if and only if one of the following conditions is satisfied:
\begin{enumerate}
\item[i)] $M$ is diagonalizable in $M_{2}(\mathbf{R})$;
\item[ii)] $\chi_{M}$ has two complex conjugate roots with non-zero imaginary part.
\end{enumerate}