I know that potential pyronuclear reactions in black dwarfs can lower their Chandrasekhar limit causing runaway fusion, but can a black dwarf undergo a normal Type Ia supernova? That is, if a fully cooled black dwarf begins accretion or collides into another black dwarf, will they undergo runaway carbon burning and explode?
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Yes. That is effectively what type Ia supernovae are thought to be.
The interiors of old carbon white dwarfs in the universe today are already "cold" enough to have crystallised. Those with masses above about 1.3 solar masses are susceptible to pycnonuclear ignition if they accrete more mass. If the accreted material is helium rich, then the ignition might take place in that material as it is compressed into the white dwarf.
Mergers are another route. The merger produces a massive white dwarf that collapses as it cools and may explode.
Both of these mechanisms don't really care how absolutely cold the progenitor white dwarf(s) is (are).
