I can not motivative the problem to be honest, but the following identity has shown up in my research. I have verified the identity for large enough values of $k$ using sagemath. Now, I want to prove it: $$\binom{2k}{k}\left[\sum_{r=0}^k\ (-1)^r\binom{k}{r}\binom{3k-r}{k-r}\binom{3k-r}{k}\right]=(-1)^k\left[\sum_{r=0}^{2k} (-1)^r\binom{2k}{r}^4\right].$$
At first I tried to evaluate the right hand side but couldn't. I could see that there is a similar identity (Dixon's identity when the exponent is $3$). I am clueless what to do with the left hand side! After everything gone, once I tried to prove the identity by mathematical induction on $k$ but as expected, it turned out to be a horrible task.
Any suggestion or comments will be highly appreciable.
Notations: $\binom{n}{r}$ is the binomial coefficient i.e. the coefficient of $x^r$ in $(1+x)^n.$
Edited: The last binomial factor should be $\binom{3k-r}{k}$ instead of $\binom{2k-r}{k}.$