say $k_1/k$ is a finite unramified extension of non-archimedean local fields, and $G/k_1$ is unramified connected reductive, and set $H={\operatorname{Res}}_{k_1/k}(G)$. Then $H$ is unramified over $k_1$, and if $\pi$ is a representation of $G(k_1)=H(k)$ then $\pi$ is unramified as a representation of $G(k_1)$ if and only if it is unramified as a representation of $H(k)$. Furthermore, the two notions of being defined over $E$ (one for $G$ and one for $H$) coincide. Moreover, the two notions of having Satake parameter defined over $E$—one for $G$ and one for $H$—also coincide; we give the argument for this as it is a little trickier. Let $T_d$ denote a maximal split torus in $G$ and let $T$ denote its centralizer. The Satake homomorphism for $G$ is an injective ring homomorphism from an unramified Hecke algebra for $G$ into ${\mathbb{C}}[T(k_1)/U]$, with $U$ a maximal compact subgroup of $T(k_1)$. The Satake homomorphism for $H$ is a map between the same two rings, and it can be easily checked from the construction in Theorem 4.1 of [@cartier:corvallis] that it is in fact the same map. The homomorphism for $G$ is an isomorphism onto the subring ${\mathbb{C}}[T(k_1)/U]^{W(G)}$ of ${\mathbb{C}}[T(k_1)/U]$, with $W(G)$ the relative Weyl group for the 