Let $A \in M_3 (\Bbb R)$ and let $X = \left \{C \in GL_3 (\Bbb R)\ |\ CAC^{-1}\ \text {is triangular} \right \}.$ Then
$(1)$ $X \neq \varnothing$
$(2)$ If $X = \varnothing,$ then $A$ is not diagonalisable over $\Bbb C$
$(3)$ If $X = \varnothing,$ then $A$ is diagonalisable over $\Bbb C$
$(4)$ If $X = \varnothing,$ then $A$ has no real eigenvalue
My attempt $:$ $(1)$, $(2)$ and $(4)$ are definitely false. For instance we may consider the matrix $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}.$ Now if $X = \varnothing$ that implies there exists at least one eigenvalue of $A$ which is non-real for otherwise we can transform $A$ to it's real Jordan canonical form which is upper triangular and hence $X \neq \varnothing,$ a contradiction to our hypothesis. How does that imply that $A$ is diagonalizable over $\Bbb C\ $? Any help in this regard will be appreciated.
Thanks for your time.