Let $G$ be a Lie group and $V$ be a vector space. Let $\rho_{l} : G \times V \to V$ be a left representation and $\rho_r:V \times G \to V$ be a right representation which commutes with $\rho_l$ in the sense that $\rho_l(g_1 , \rho_r(v , g_2) ) = \rho_r( \rho_l(g_1, v) , g_2)$. Then the group multiplication $(g_1,v_1) \cdot (g_2,v_2) = (g_1 \cdot g_2 \quad,\quad \rho_r(v_1,g_2) + \rho_l(g_1,v_2))$ makes the set $G \times V$ into a Lie group. The identity is $(e_G, e_H)$ and the inverse of $(g,h)$ is $(g^{1},  \rho_l[ g^{1} , \rho_r(h,g^{1} ) ] )$ and associativity can be verified by hand. This Lie group appears to be some sort of sum of a left semidirect product with a right semidirect product. Does it have a name?
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It is isomorphic to the semidirect product of $G$ with $V$ for the left representation $g\mapsto (v\mapsto g.v.g^{1})$. 

