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A flag manifold of a compact semisimple Lie group $G$ is defined as a quotient $M=G/K$ where $K$ is the centralizer of a one-parameter subgroup $exp\left(tx\right)$ of $G$. Then $M$ can be identified with the adjoint orbit of $x$ in the Lie algebra $𝒢$ of $G$. Two flag manifolds $M=G/K$ and $M^{\text{'}}=G/K^{\text{'}}$ are equivalent if there exists an automorphism $\phi :G\to G$ such that $\phi \left(K\right)=K^{\text{'}}$ (equivalent manifolds need not be $G$-diffeomorphic since $\phi$ is not assumed to be inner). In this article, explicit formulas for decompositions of the isotropy representation for all flag manifolds...