The ternary quadratic form $Q(x,y,z) = -3 x^2 + 3 x y + 3 x z - y^2 - y z - z^2$ is not positive definite. In Sage, using code found in this link,
Q = TernaryQF([-3, -1, -1, -1, 3, 3])
Qr, m = Q.reduced_form_eisenstein()
Qr
Qr.is_eisenstein_reduced()
nothing is returned by Sage. Why is nothing returned by Sage, what are the conditions on using Eisenstein reduction, what is a reference on Eisenstein reduction of ternary quadratic forms, and if $Q(x, y, z)$ does not satisfy the conditions for Eisenstein reduction, is there another appropriate reduction that might be used in such a case?
your form is negative semidefinite, a binary form in disguise. Note that the determinant of the Hessian matrix of second partials is zero.
In my standard order $$ x^2, y^2, z^2, yz, zx, xy $$ you have ordered coefficients $$ -3,-1,-1,-1,3,3 $$ This is $SL_3 \mathbb Z$ equivalent to $$ -1,-1,-1,1,1,1 $$ or $$ -\frac{1}{2} \left( (v-w)^2 + (w-u)^2 + (u-v)^2 \right) $$ by
$$ x=u, \; y = u+v-w , z=w $$
This is also $$ - \frac{1}{4} (2u - v - w )^2 - \frac{3}{4} ( v - w )^2 $$ of deficient rank