Abstract
This work is the second in a series, following Part I [Kwa24] (
Algebra Number Theory
18
.10 (2024)) and preceding Part III [Kwa25] (
Math. Ann.
391
.1 (2025)). We continue our investigation of spectral moments of
GL
(
3
)
×
GL
(
2
)
L
$\text {GL}(3)\times \text {GL}(2)\ L$
GL left parenthesis 3 right parenthesis times GL left parenthesis 2 right parenthesis upper L
-functions from the perspective of period integrals. Using an identity between two distinct periods for the
GL
(
3
)
$\text {GL}(3)$
GL left parenthesis 3 right parenthesis
Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of
GL
(
2
)
L
$\text {GL}(2)\ L$
GL left parenthesis 2 right parenthesis upper L
-functions to the shifted fourth moment of
GL
(
1
)
L
$\text {GL}(1) L$
GL left parenthesis 1 right parenthesis upper L
-functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures (
Proc. Lond. Math. Soc.
(3)
91
(2005)).