All the asset register is, is a book listing all of the assets together with their original cost, the accumulated depreciation, and the net book value (carrying value).
It is not part of the double entry – just a book (to help the business keep track of their non-current assets – many businesses stick a number on all of their assets just so that they can keep control).
Again, it is not part of the double entry, but it does act as a check because the balances in the asset account (at cost) and the accumulated depreciation account should agree with the totals in the asset register.