Let us consider
Brannen's circulant sum form for the
tribimaximal mixing matrix. There are a few ways to do this, but we will start by mixing up $1$-circulant and $2$-circulant parts to obtain:
Note that the $2 \times 2$ block is not $F_2$, but rather the MUB
alternative $R_2$. This gives a purely two dimensional product form for the tribimaximal matrix:
in terms of $R_2$ type operators, with $4$ parameters $A$, $B$, $X$ and $Y$. Note that it doesn't matter much if we swap $\sqrt{2}^{-1}$ for $\sqrt{2}$. There are alternative forms of this result, again making the tribimaximal matrix robust.
Note that the $2 \times 2$ block is not $F_2$, but rather the MUB alternative $R_2$. This gives a purely two dimensional product form for the tribimaximal matrix: 
in terms of $R_2$ type operators, with $4$ parameters $A$, $B$, $X$ and $Y$. Note that it doesn't matter much if we swap $\sqrt{2}^{-1}$ for $\sqrt{2}$. There are alternative forms of this result, again making the tribimaximal matrix robust.
And note that the triple product of three such doubly magic matrices (one for each dimension) gives a neat parameterisation of all such (eg. CKM) matrices.
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