On the standard conjecture for complex 4-dimensional elliptic varieties

We prove that the Grothendieck standard conjecture $B(X)$ of Lefschetz type on the algebraicity of operators $\ast$ and $\Lambda$ of Hodge theory holds for every smooth complex projective model $X$ of the fibre product $X_1\times_C X_2$, where $X_1\to C$ is an elliptic surface over a smooth projective curve $C$ and $X_2\to C$ is a morphism of a smooth projective threefold onto $C$ such that one of the following conditions holds: a generic geometric fibre $X_{2s}$ is an Enriques surface; all fibres of the morphism $X_2\to C$ are smooth $\mathrm{K}3$-surfaces and the Hodge group $\operatorname{Hg}(X_{2s})$ of the generic geometric fibre $X_{2s}$ has no geometric simple factors of type $A_1$ (the assumption on the Hodge group holds automatically if the number $22-\operatorname{rank}\operatorname{NS}(X_{2s})$ is not divisible by 4).