Field made of real&imaginary parts of members of another field

Question

Let ${\mathbb K}$ be a normal extension of $\mathbb Q$. Let $\mathbb L$ be the subfield of ${\mathbb C}$ generated by the real and imaginary parts of elements of $\mathbb K$. Thus $\mathbb L$ is a subfield of $\mathbb R$. Must $\mathbb L$ be a normal extension of $\mathbb Q$ also ?