$A,B,C,D\in M_n(F)$. $AB^T$ and $CD^T$ are two symmetric matrices, show if $AD^T-BC^T=I$, then $A^TD-C^TB=I$

Question

$M_n(F)$ is the set of all $n\times n$ matrices over field $F$. Assume $A,B,C,D\in M_n(F)$. $AB^T$ and $CD^T$ are two symmetric matrices. Show if $AD^T-BC^T=I$, then $A^TD-C^TB=I$.

I know about these:

• elementary operations
• row-reduced echelon forms
• linear dependence or independence
• equivalent matrices
• matrix block multiplication

My attempt: I used symmetric matrices definitions: $AB^T=BA^T$ and $CD^T=DC^T$. Then I tired to build those terms by multiplying these matrices, but I couldn't reach any thing useful. Even hints or giving ideas are appreciated! Thanks.