 Exponential criterion for a continuous local martingale and its bracket

Suppose $X,A$ are continuous, start at zero, and $A$ is increasing. If $e^{X-A/2}$ and $e^{-X-A/2}$ are local martingales, their logarithms first recover adaptation of $X,A$ and the semimartingale property of $X$. In the decomposition $X=N+V$, the two Itô drift measures give $dV+(d[N]-dA)/2=0$ and $-dV+(d[N]-dA)/2=0$. Thus $V=0$ and $[X]=A$. All real exponential parameters may be assumed, but these two suffice.