an element $g\in G$, then it holds for all elements in the double coset $HgH$. By this, to verify the “SCF” condition of the pair $H\subset G$ it suffices to check the condition (\[Eq2\]) for a set of representatives of the double coset space $H\backslash G/H$.

Symmetric pairs
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Let $G$ be a connected compact Lie group. Endow $G$ with a biinvariant Riemannian metric. Particularly it gives a $G$ conjugation invariant positive-definite inner product on ${\mathfrak{g}}_0$. Write ${\mathfrak{p}}_0$ for the orthogonal complement of ${\mathfrak{h}}_0$ in ${\mathfrak{g}}_0$.

\[L:symmetric1\] We have $G=H^{0}\exp({\mathfrak{p}}_0)$.

For any $g\in G$. Choose a minimal length geodesic $\gamma:[0,1]\rightarrow G$ connecting $H^{0}$ and $H^{0}g$. By the biinvariance of the metric, we may assume that $\gamma(0)=1\in H^0$. Then, there exists $X\in{\mathfrak{g}}_0$ such that$\gamma(t)=\exp(tX)$ ($\forall t\in[0,1]$). Due to $\gamma$ is of minimal length, we have $X\perp
{\mathfrak{h}}_0$. That means, $X\in{\mathfrak{p}}_0$. Write $\gamma(1)=xg$ for some $x\in H^{0}$. Then, $$g=x^{-1}\gamma(1)=x^{-1}\exp(X)\in H^{0}\exp({\mathfrak{p}}_0).$$ This shows the conclusion.

In case $H\subset G$ is a symmetric pair, Lemma \[L:symmetric1\] could be further simplified. Let $G$ be a connected compact Lie group and $\theta\in\Aut(G)$ be an involutive automorphism. Write $H=G^{\theta}$. Choose a maximal abelian subspace ${\mathfrak{a}}_0$ of ${\mathfrak{p}}_0$, and set $A=\exp({\mathfrak{a}}_0)$.

\[L:symmetric2\] We have $G=H^{0}AH^{0}$.

By [@Knapp Proposition 