Finite-dimensional complex representations of nilpotent Lie algebras over a subfield of $\Bbb{C}$

Question

From Lie's theorem we know that the complex representations of any solvable Lie algebra $\mathfrak{g}$ over a subfield of $\Bbb{C}$ are such that there exists a basis in which the representation matrices are upper triangular. Nilpotent Lie algebras are solvable, so Lie's theorem applies to them as well. Algebras of upper diagonal (finite-dimensional) matrices with zero diagonal entries are nilpotent Lie algebras (with the commutator bracket); moreover, from Engel's theorem, we know that the adjoint representation of a nilpotent Lie algebra admits a basis in which the representation matrices are upper triangular with zero diagonal entries. But is it true that for the representations of any nilpotent Lie algebra over a subfield of $\Bbb{C}$ there exists a basis in which the representation matrices are upper triangular with zero diagonal entries complex representations?