Crystalline lifts of semisimple $G$-valued Galois representations with fixed determinant

For a finite extension $K\mathrm{/}{\mathbb{Q}}_p$ and a split reductive group $G$ over ${\mathcal{O}}_K$, let $\bar{\rho }\text{\hspace{0.17em}⁣}:{\mathrm{G}\mathrm{a}\mathrm{l}}_K\to G({\bar{\mathbb{F}}}_p)$ be a continuous quasi-semisimple mod $p$ $G$-valued representation of the absolute Galois group ${\mathrm{G}\mathrm{a}\mathrm{l}}_K$. Let ${\bar{\rho }}^{\mathrm{a}\mathrm{b}}$ be the abelianization of $\bar{\rho }$ and fix a crystalline lift $\psi$ of ${\bar{\rho }}^{\mathrm{a}\mathrm{b}}$. We show the existence of a crystalline lift $\rho$ of $\bar{\rho }$ with regular Hodge-Tate weights such that the abelianization of $\rho$ coincides with $\psi$. We also show analogous results in the case that $G$ is a quasi-split tame group and $\bar{\rho }\text{\hspace{0.17em}⁣}:{\mathrm{G}\mathrm{a}\mathrm{l}}_K\to {}^{L}G({\bar{\mathbb{F}}}_p)$ is a semisimple mod $p$ $L$-parameter. These theorems are generalizations of those of Lin and Böckle-Iyengar-Paškūnas.