Similarity of a real matrix

Question

If $S$ is a triangularizable matrix in $M_{n}(\mathbb{R})$ such that $\rho{(S)}=1$(the spectral radius). Can we get that $S$ is similar to $$ \begin{bmatrix} B & 0 \\ 0 & C \end{bmatrix}$$ where spectrum of $B$ is on unit circle and spectrum of $C$ is into unit circle?

Answer 1

No. The matrix

$$ S = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $$

has $\rho(S) = 1$ but it is not similar to a non-trivial block diagonal matrix because it is not diagonalizable.

Alternatively, consider

$$ S = \begin{pmatrix} 1 & 0 \\ 0 & \frac{1}{2} \end{pmatrix} $$

which has $\rho(S) = 1$ but is not similar to a block diagonal matrix (trivial or not) for which the blocks have a spectrum that lie on the unit circle.