  For a linear algebraic group $G$ over $\bf Q$, we consider the period domains
$D$ classifying $G$-mixed Hodge structures, and construct the extended period
domains $D_{\mathrm{BS}}$, $D_{\mathrm{SL}(2)}$, and $\Gamma \backslash
D_{\Sigma}$. In particular, we give toroidal partial compactifications of mixed
Mumford--Tate domains.
