Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 3 / 3 / c / ii

For a block-diagonal element, form preservation saysGiven any $C\in \mathrm{GL}(W)$, the first equation uniquely determinesWith dual bases ordered in matching rather than reversed order, the same relation is simply $A=C^{-T}$. This is the contragredient action on the paired space $W^{\prime }$.

The second equation independently allows every $B\in \mathrm{Sp}(U)$. The map taking a block-diagonal element to $(C,B)$ is a group isomorphism, with inverse $(C,B)\mapsto (R{C}^{-T}R,B,C,0,0,0)$. Thus