In Jacobson's Lie Algebras, page 303, it seems he uses the following result:
If $\mathfrak L$ is a simple finite-dimensional Lie algebra over a field $\Omega$ which is the algebraic closure of a field $\Phi$, then there exists a basis for $\mathfrak L$ whose multiplication coefficients are in $\Phi$.
Is this theorem true as stated, and can someone give me a reference to a proof in some textbook? Preferably in Jacobson's books or in Lang's Algebra, if possible.
On page $313$ Jacobson gives the following reasoning ($4$ lines before lemma $3$):
"$L$ is simple over the algebraically closed field $\Omega$. Hence it has a basis over $\Omega$ whose multiplication coefficients are in the prime field and so are in $\Phi$."
That the coefficients are in the prime field follows from the classification. There is even a basis (a so-called Chevalley basis) such that the coefficients are in $\mathbb{Z}$. Tensoring this $\mathbb{Z}$-Lie algebra with any field $F$ gives a Lie algebra over $F$.
For more details on a Chevalley basis and the reduction modulo a prime $p$ see Chapter VII, section $25$ in the book Introduction to Lie Algebras and Representation Theory by James E. Humphreys.