Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity:$$J_{ij,k\ell} := \operatorname{Im}(u_{ik} u_{j\ell} u_{i\ell}^* u_{jk}^*)$$where $z^*$ denotes the complex conjugate of $z$. Clearly, $J_{ij,k\ell}$ is antisymmetric under exchange of $i$ with $j$ or under exchange of $k$ with $\ell$, and symmetric under exchange of $(i,j)$ with $(k,\ell)$; but, less obviously, it is also invariant if $i$ and $j$ are translated mod $3$, or if $k$ and $\ell$ are. So in fact, there is just one such quantity (up to an arbitrary sign convention), say $J := J_{12,12}$. This is the Jarlskog invariant from particle physics (used in the description of the unitary Cabibbo-Kobayashi-Maskawa matrix). Its importance stems from the fact that it is clearly invariant under multiplication of the rows, or of the columns, of $U$, by arbitrary complex numbers of modulus $1$ (i.e., it is an invariant of $\mathit{U}_1^3\backslash\mathit{U}_3/\mathit{U}_1^3$).
This is just an easy computation, but I am left wondering where this invariant "comes from", conceptually:
Q1: Is there a more sophisticated way to describe this invariant (and perhaps generalize it to $n\times n$ unitary matrices)?
To better orient the question in the $3\times 3$ case, maybe I should reframe it as follows:
Consider three points $A,B,C$ in $\mathbb{P}^2(\mathbb{C})$ at Fubini-Study distance $\pi/2$ from one another, meaning that they lift to orthogonal vectors in $\mathbb{C}^3$ (for the usual Hermitian product on $\mathbb{C}^3$), call this a "self-polar triple"; and consider another self-polar triple $A',B',C'$. Now there exists a unitary matrix $U$ taking $A,B,C$ to $A',B',C'$ respectively (by normalizing each point to a vector of unit norm in $\mathbb{C}^3$), and this matrix is uniquely defined up to multiplication of its rows, or of its columns, by arbitrary complex numbers of modulus $1$. So it makes sense to consider the Jarlskog invariant of $U$ as an invariant of the position of the triple $(A',B',C')$ relative to $(A,B,C)$. What does this invariant tell us, geometrically?
Q2: How can we reformulate the Jarlskog invariant of two self-polar triples $(A,B,C)$ and $(A',B',C')$ as above in terms of the geometry of $\mathbb{P}^2(\mathbb{C})$ (e.g., distances, Euclidean/Hermitian/Kähler angles, shape invariants of triangles, etc.)?
(For example, if I am not mistaken, the vanishing of the Jarlskog invariant tells us precisely that $A,B,C$ and $A',B',C'$ lie on a common $\mathbb{P}^2(\mathbb{R})$ inside $\mathbb{P}^2(\mathbb{C})$.)
Xref:This physics.stackexchange question is related but does not really answer my conceptual question (also, the link given in the answer is broken and I was unable to find the resource online).
