I had a Triumph with a meta alarm fitted via the ignition switch and when I had trouble with that it did not show up anywhere
Most triumphs have a dedicated loom plug for alarms and on modern triumphs it taps into the fuel pump circuit and older carb models I think it's the kill switch circuit.
On budget models and some deluxe models triumph seem to scrimp on their electronics and make them pretty basic.
The dedicated loom plug for the alarm is completely separate from the FI system so when an alarm goes wrong it's won't recognise it leaving the owner clueless.
Older DL's have a safety system that throws up "check" on the dash if the side stand is down ,the kill switch is off and the clutch switch (I think).
If the immobiliser part of the alarm is faulty and it's fitted to the ignition or the kill switch it should throw up "check" on the dash as the ignition circuit has been cut off somehow.
Most alarms are fitted this way and if it's faulty it will leave an open circuit on the ignition.
Of course, I'm just assuming the alarm is faulty and it could be something else .? but, as the alarm is ancient it's more than likely that its on its way out. Just my opinion, I maybe wrong.
If so, have it taken out, they are more agro than they are worth.