To find the Lie algebra of the Heisenberg group $H$, which we know to consist of upper triangular matrices, we see that exponentials of all strictly upper triangular matrices are in $H$. I do not get the following could you explain?
"On the other hand if $X$ is any matrix such that $e^{tX}$ is upper triangular, then all the entries of $X=\frac{d}{dt}|_{t=0}e^{tX}$ which are on or below diagonal must be zero so that $X$ is upper triangular"
Thus the Lie algebra of the Heisenberg group is the space of all $3\times 3$ real matrices which are strictly upper triangular.