  Recently, we determined the detailed energy level structure of the
$X^1\Sigma^+$, $A^1\Pi$ and $a^3\Pi$ states of AlF that are relevant to laser
cooling and trapping experiments. Here, we investigate the $b^3\Sigma^+, v=0$
state of the AlF molecule. A rotationally-resolved (1+2)-REMPI spectrum of the
$b^3\Sigma^+, v'=0 \leftarrow a^3\Pi, v''=0$ band is presented and the lifetime
of the $b^3\Sigma^+, v=0$ state is measured to be 190(2)~ns.
Hyperfine-resolved, laser-induced fluorescence spectra of the $b^3\Sigma^+,
v'=0 \leftarrow X^1\Sigma^+, v''=1$ and the $b^3\Sigma^+, v'=0 \leftarrow
a^3\Pi, v''=0$ bands are recorded to determine fine- and hyperfine structure
parameters. The interaction between the $b^3\Sigma^+, v=0$ and the nearby
$A^1\Pi$ state is studied and the magnitude of the spin-orbit coupling between
the two electronic states is derived using three independent methods to give a
consistent value of 10(1)~cm$^{-1}$. The triplet character of the $A$ state
causes an $A\rightarrow a$ loss from the main $A-X$ laser cooling cycle below
the 10$^{-6}$ level.
