Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be an good, simple lower bound for $G(F_q)$?

One can get something fairly nice from a general lower bound on the number of points on a variety over $\mathbb{F}_q$, in the style of Lang-Weil (Cafure-Matera 2006, Thm 7.5, is the best result I know). However - perhaps one can do something better for a group?

Example: for a Chevalley group, we have $|G(\mathbb{F}_q)|\geq q^d - d q^{d-1}$. Perhaps something like that is also true in general?

Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$?

One can get something fairly nice from a general lower bound on the number of points on a variety over $\mathbb{F}_q$, in the style of Lang-Weil (Cafure-Matera 2006, Thm 7.5, is the best result I know). However - perhaps one can do something better for a group?

Example: for a Chevalley group, we have $|G(\mathbb{F}_q)|\geq q^d - d q^{d-1}$. Perhaps something like that is also true in general?