Statement

The process

$$ B_t^2-t $$

is a martingale.

Verification

For $s $$ B_t=B_s+(B_t-B_s). $$

Expanding and conditioning on $\mathcal F_s$ gives

$$ \mathbb E[B_t^2-t\mid \mathcal F_s]

B_s^2+2B_s\mathbb E[B_t-B_s]+ \mathbb E[(B_t-B_s)^2]-t. $$

Because

$$ \mathbb E[B_t-B_s]=0, \qquad \mathbb E[(B_t-B_s)^2]=t-s, $$

we get

$$ \mathbb E[B_t^2-t\mid \mathcal F_s]=B_s^2-s. $$