The set of integers $n$ expressible as $n=x^2+xy+y^2$ 
Let $S$ be the set of integers $n$, such there exist integers $x,y$ with $$n=x^2+xy+y^2$$


Is the implication $$a,b\in S\implies ab\in S$$ true? If yes, how can I prove it?

I worked out $$n\in S\iff 4n\in S$$ and $$n\in S\iff 3n\in S$$
I tried two approaches. The first is to express $$(a^2+ab+b^2)(c^2+cd+d^2)$$ in the form $$f^2+fg+g^2$$ with polynomials $f,g$ with integer coefficients. I however could not find suitable $f$ and $g$.
The second approach is based on $$x^2+xy+y^2=\frac{(2x+y)^2+3y^2}{4}$$ If we have $n=x^2+xy+y^2$ , we have $u^2+3v^2=4n$ for some integers $u,v$ with equal parity. The main problem of this approach is to consider the equal parity.
Any ideas ?

Note that
\begin{align*}
&\qquad16(a^2+ab+b^2)(c^2+cd+d^2)\\&=\left((2a+b)^2+3b^2\right)\left((2c+d)^2+3d^2\right)\\&=\left(\frac{(2a+b)(2c+d)+3bd}{2}\right)^2+3\left(\frac{d(2a+b)-b(2c+d)}{2}\right)^2
\end{align*}
