This answer is just to record that Will Sawin's example works: The Borel subgroup of $PGL_2$ is not a complete intersection in $PGL_2$. Recall that the coordinate ring of $GL_2$ is $k[w,x,y,z,\Delta^{-1}]$ where $\Delta=wz-xy$. We are thinking of $w$, $x$, $y$, $z$ as entries of the matrix $\left( \begin{smallmatrix} w & x \\ y & z \end{smallmatrix} \right)$. The coordinate ring of $PGL_2$ is the subring of homogenous elements, where $w$, $x$, $y$ and $z$ are graded in degree $1$ and hence $\Delta^{-1}$ has degree $-2$.
The Borel is given by the equations $wy/\Delta=xy/\Delta=y^2/\Delta=yz/\Delta=0$. Conceptually, the defining equation is $y=0$, but the element $y$ isn't in the coordinate ring of $PGL_2$. Since the Borel is a hypersurface, if it were a complete intersection, it would be defined by the vanishing of a single equation, say $g(w,x,y,z)/\Delta^k$, where $g$ is a homogeneous polynomial of degree $2k$. We may also assume that $\Delta$ does not divide $g$. Since $g$ has even degree, it must either vanish to even order along $y=0$, or else vanish along some other hypersurface besides $y=0$. Either way, $g/\Delta^k$ does not define the ideal of the Borel.
