The Lie algebra quotient of a maximal ideal

Question

If $\mathcal g$ is a Lie algebra, and $Z(\mathcal g)$ is its centralizer. Suppose that $Z(\mathcal g)$ is a maximal ideal of $\mathcal g$. Is it correct that $\mathcal g/Z(\mathcal g)$ is of dimension $1$? How to prove that? If $I$ is any maximal ideal, is the statement above still correct?