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Let $G$ be a compact Lie group and $H, H' \leq G$ Lie subgroups. Consider the set $M = H' \cdot H = \{h\cdot h' \ \vert \ h \in H, h' \in H'\}$. Is it possible to describe explicitly the tangent space of $M$ at any point $x \in M$ in terms of tangent spaces of $H$ and $H'$, i.e. their Lie algebras $\mathfrak{h}, \mathfrak{h'}$? Since $M$ is not a group, one cannot simply take the Lie algebra generated by $\mathfrak{h}, \mathfrak{h'}$.

Added: can this be generalized to sets of the form $H_1 \cdot \dots \cdot H_n$?

The concrete example I'm interested is the unitary group $G = U(2^N)$ acting on $(\mathbb{C^2})^{\otimes n}$ and $H, H'$ being unitary groups acting on only $2$ different components (say 1-2 and 2-3: $H = U(4) \otimes I \dots\otimes I, H' = I \otimes U(4) \otimes I \dots \otimes I$).

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Your $M$ is an homogeneous space for a (left) action of $H'\times H^{\mathrm{op}}$. So you can do what one usually does in that situation. –  Mariano Suárez-Alvarez Nov 19 '12 at 3:38
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