Not being diagonalisable condition

Question

Let $$A = \begin{pmatrix} -1 & 2 & -2 \\ 5 & -1 & 6 \\ 6 & -2 & 7 \end{pmatrix}$$.
The matrix $A$ has eigenvalues $5, \pm i$. Now I know that $A$ is only diagonalisable over $\mathbb{C}$, and not diagonalisable over $\mathbb{R}$, since we only have one real eigenvalue. But it says that $A$ is not diagonalisable, just because it has one eigenvalue. Isn't this the sole reason?
Since the eigenspaces over $\mathbb{R}$ only have one dimension, then it's not diagonalisable.