Holme type theorem for special linear groups

Let Z be an affine algebraic variety and ED(Z)= max(2 dim Z+1, dim TZ). Let X be a smooth algebraic variety isomorphic to a semi-simple linear algebraic group whose Lie algebra is a sum of special linear Lie algebras. We show that if dim X > ED(Z) -1, then Z admits a closed embedding into X. We also show that for every smooth affine flexible variety Y there is a closed embedding of $Z$ into the the product of Y and an affine n-space provided that n > dim Z- 2 and dim Y +n > ED (Z) -1.