Let $A \in M_n(\mathbb{R})$ be a matrix which is diagonalizable in $M_n(\mathbb{C})$. We note $u_{A, \mathbb{R}}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ the endomorphism defined by the left multiplication by A. Show that $\mathbb{R}^n$ is a direct sum of lines and planes invariant by $u_{A, \mathbb{R}}$, the restriction of $u_{A, \mathbb{R}}$ to each invariant plane being given, in an appropriate basis, by a matrix of the form $$ S_{a, b}=\left(\begin{array}{cc} a & b \\ -b & a \end{array}\right) $$
If $A$ is diagonalizable in $M_n(\mathbb{C})$ but not in $M_n(\mathbb{R})$, we know that $A$ has complex eigenvalues. If we consider the endomorphism $u_{A, \mathbb{C}}$ and a complex eigenvalue $\mu$ and the associated vector $v \in \mathbb{C^n}$ of $u_{A, \mathbb{C}}$ then we know that the plane $P = (h_1, h_2)$ where $h_1= \text{Re}(v)$ and $h_2 = \text{Im}(v) $ is invariant under $u_{A, \mathbb{R}}$. I don't know where to go from here. Is the restriction of $u_{A, \mathbb{R}}$ to the plane $P$ the matrix $S_{a,b}$?
Planes correspond to complex eigenvalues (which come in conjugate pairs) Since the matrix is diagonalizable, it has the property that whenever it has an invariant subspace, it has a supplementary invariant subspace (such endomorphisms are called semisimple), Now as u mentioned the plane $P$ is invariant, verify that the restriction of A to $P$ has the form mentioned in the exercise. Now u can proceed by induction on the dimension of the space, the result being trivial for $n=1$,suppose it true for every $k\leq n$, let $A$ be an endomorphism of $\mathbb{R} ^{n+1} $,Since $A$ is diagonalizable, it has an eigenvalue, hence an invariant subspace , if this eigenvalue is real, the subspace contains a line, if it is complex, the subspace contains a plane, and in either cas, there's a supplementary also inveriant, now this supplémentary has either dimension $n$ or $n-1$,in either case the induction hypothesis applies, to prove that this supplementary decomposes as the sum of lines and planes, hence $\mathbb{R} ^{n+1} $ decomposes as the sum of lines and planes. In fact one can start with the following :Let $A\in M_{2}(\mathbb{R} $ be st its caracterestic polynomial has complex(conjugate) eigenvalues, then $A$ is conjugate to a similarity(a rotation, composed with a homotethy) which is exactly $S{a, b} $