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For every integer $n\\geq 1$, we consider the generalized Galois symbol $K(k;G_1,G_2)/n\\xrightarrow{s_n} H^2(k,G_1[n]\\otimes G_2[n])$, where $k$ is a finite extension of $\\mathbb{Q}_p$, $G_1,G_2$ are semi-abelian varieties over $k$ and $K(k;G_1,G_2)$ is the Somekawa K-group attached to $G_1, G_2$. Under some mild assumptions, we describe the exact annihilator of the image of $s_n$ under the Tate duality perfect pairing, $H^2(k,G_1[n]\\otimes G_2[n])\\times H^0(k,Hom(G_1[n]\\otimes G_2[n],\\mu_n))\\rightarrow\\mathbb{Z}/n$. 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