Can I use 1.5mm2 1 core 7/0.5mm2 strand building wire rated for .6/1kV AC in small <24V DC applications.
TL;DR While you should be able to use it based on voltage alone, there may be reasons why you shouldn't.
Cable has several specifications to meet, voltage related (insulation), current related (area and copper purity), mechanical (insulation thickness, robustness, flexibility, and copper strand size and number). There are others, for instance insulation softening temperature, flammability and whether it releases toxic gases when it burns.
It's fine to use a cable at a lower voltage than rated. For small diameter cables, there's no real difference between DC and low frequency AC.
BTW, your cable area and build has an inconsistent specification, if it's 7/0.5mm, then the total copper area is 1.37mm2. If it's 7/0.5mm2, then the total is 3.5mm2. So it's probably the former, and they've helpfully 'rounded up'.
7/0.5mm is a curious configuration. The strands seem to be too thick for a true 'flexible' power cord, but thinner than they need to be for stapled-down power cable. It's probably intended as equipment wire for the internal cabling of cabinets, and not for continuous flexing.
With high voltage systems, cables tend to be rated thermally for current only as the voltage drop is usually an insignificant fraction of the total voltage available. With low voltage systems like 24V, you usually need to take account of the voltage drop as well as the thermal limit for the cable, often ending up with a thicker cable than you expected.
small <24VDC application is not a complete specification. If it's automotive, then using wire not rated for automotive use may be a non-approved 'modification' and invalidate your insurance, depending on your location. If it releases toxic gases on burning, or doesn't remain flexible down to -20 degrees, then it could cause dangers not forseen by just staying within the voltage and current rating.