linear computations over bilinear pairings Does this hold in asymetric bilinear pairings?

$e(x_1,x_2)e(x_3,x_4) = e(x_1x_3,x_2x_4)$, where $x_1,x_3 \in \mathbb{G}_1$ and $x_2,x_4 \in \mathbb{G}_2$ for a bilinear pairing $e$
For clarity, I will use an additive notation in the groups $\mathbb{G}_1$ and $\mathbb{G}_2$, and denote group elements with the "$x$" letter, not "$e$".

For all $x_1, x_3 \in \mathbb{G}_1$ and $x_2, x_4 \in \mathbb{G}_2$, we have the following:

\begin{eqnarray*}
e(x_1 + x_3, x_2 + x_4) &amp;=&amp; e(x_1, x_2 + x_4) e(x_3, x_2 + x_4) \\
&amp;=&amp; e(x_1, x_2) e(x_1, x_4) e(x_3, x_2) e(x_3, x_4)
\end{eqnarray*}

(First equation is by linearity of the pairing for its first operand, second equation is by linearity for its second operand.)

Therefore, one can have the property you seek only when $e(x_1,x_4)$ and $e(x_3,x_2)$ happen to be opposite of each other, which is not true in general (unless the pairing is degenerate, in which case it is quite uninteresting).