In his paper Invariant Kahler metrics and projective embeddings of the flag manifold, Bull. Austal. Math. Soc. 49 (1994), K. Yang considers the flag manifold $$F_{1,2,3}(\mathbb{C}^3):=SU(3)/S(U(1)^3)$$ and determines the space of invariant Hermitian and inveriant Kahler metrics on it.
In particular, he shows that a Killing metric is not Kahler.
Moreover
On the other hand, by applying Kodaira embedding theorem, he deduces proves that $F_{1,2,3}$ F_{1,2,3}(\mathbb{C}^3)$ is projective algebraic and provides an explicit projective embedding of it.
The computations are made quite explicitly in terms of the Maurer-Cartan form of $SU(3)$.
So this could be one of the examples you are looking for.
