NIST implementation of the Lucas primality test
Thanks to the references provided by @fgrieu I found the relevant equation in "New
Primality Criteria and Factorizations of2^m+-1" by Brillhart-Lehmer-Selfridge.

In there, Equation (14) says:

$2V_{r+s} = V_rV_s+DU_rU_s$

Which is trivial to demonstrate, given that $D=(\alpha - \beta)^2$.