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.
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. 1 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 Hermitian and inveriant Kahler metrics on it. In particular, he shows that a Killing metric is not Kahler. Moreover, by applying Kodaira embedding theorem he deduces that$F_{1,2,3}$is projective algebraic. The computations are made quite explicitly in terms of the Maurer-Cartan form of$SU(3)\$.