Complex eigenpairs of a real matrix

Let $A \in M_n(\mathbb{R})$ be an $n \times n$ real matrix, $n \ge 2$. Consider the following two statements: (I) If $\lambda \in \mathbb{C}$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue. (II) If $v = (v_1, v_2, \dots, v_n) \in \mathbb{C}^n$ is an eigenvector corresponding to eigenvalue $\lambda = x + iy$, $y \ne 0$, then $\operatorname{Re}(v) = (\operatorname{Re}(v_1), \dots, \operatorname{Re}(v_n))$ and $\operatorname{Im}(v) = (\operatorname{Im}(v_1), \dots, \operatorname{Im}(v_n))$ are linearly independent vectors over $\mathbb{R}$. Which of the following statements is correct?