  Let $K/\mathbb{Q}_p$ be a finite extension. For all irreducible
representations $\bar\rho: G_K \to G(\bar{\mathbb{F}}_p)$ valued in a general
reductive group $G$, we construct crystalline lifts of $\bar\rho$ which are
Hodge-Tate regular. We also discuss rationality questions. We prove a variant
of the Steinberg-Winter theorem along the way.
