Let $B_3$ be the group defined by $$B_3=\langle \sigma_1,\sigma_2\mid\sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2\rangle\text{.}$$
I need to prove that the element $$g=\sigma_1^{-1}\sigma_2^{-2}\sigma_1^{-2}\sigma_2^2\sigma_1^2\sigma_2^{-2}\sigma_1^2\sigma_2^2\sigma_1^{-1}$$ is not trivial in $B_3$.
I have tried to prove the image of $g$ is not trivial in the natrual quotient homomorphism $B_3\to B_3/\langle\sigma_1^3=1\rangle$; but am not sure how to complete the argument, or should I use this approach at all. Any help will be greatly appreciated.
The following image is for those who need a geometric meaning of this question; but I don't know if it is relevant.
