Pairing notation seems to suggest that bilinear pairings could be related to Clifford Algebra (ie: Geometric Algebra); and we only have an odd choice of notation that hides this fact. For example, if EC groups $G_1$ and $G_2$ are akin to vectors, then the target group seems akin to $G_1 G_2 = G_{12} = -G_{21}$. Searches on Clifford Algebra and Elliptic Curves indicates that it might be the case; but the results are unreadable to me.
$a G_1 * (b G_2 + c G_2) = a(b+c) G_1 G_2$
I ask, because creating a CPABE (key derivation via boolean combination of attributes, where the $\land$ operation is where the problem lies) seems to be asking for a "trilinear hash" to be formed as simply as possible, like this... which atomically swaps out the user watermark for the file watermark on attributes that must all come from the same user to stop privilege escalation via collusion:
$user\ G_2\ *\ file\ G_1\ *\ (\frac{attr_0}{user} + \frac{attr_1}{user}) G_{123} = file\ *\ (attr_0 + attr_1) G_3$
In fact, the $\hat e(a G_1, b G_2)$ notation is kind of suggestive of the fact that $G_{12}$ is a bivector representing rotation from $G_1$ vector to $G_2$ vector; where $\vec u *\vec v = |\vec u| |\vec v|e^{angle_{\vec u,\vec v}}$.
Can a pairing be represented in a straight-forward way, as Geometric Algebra on Elliptic Curves? Or even just somehow with Finite Fields? Is it really one field that more rightly looks like: $a e_1 (b e_2 + c_{e2}) G$ ?
