Suppose that $X \in M_{n\times n}(\mathbb R)$, then there exists a nonsingular $S \in M_{n \times n}(\mathbb R)$ s.t $$S^{-1}XS= \begin{pmatrix} P_{1} & & & \\ &P_{2} &\ & \bigstar \\ & & \ddots & \\ & & & P_{m} \\ \end{pmatrix}$$ is block upper triangular, where each $P_i \in M_1(\mathbb R)$ or of the form $\begin{pmatrix} a &b \\ -a & b \end{pmatrix} \in M_2(\mathbb R)$ for some $b>0$.
My intuition tells me that the Schurs Triangularization theorem will do the thing but I don't know how to use the said theorem or is there any method that will solve this.
Any hints and help would be much very appreciated.