Identify compounds A and B:

Organocuprate reagent (Gilman reagent) is the coupling reagent as they couple the two carbon groups together.

Organocuprate reagent isless basic than organomagnesium and organolithium reagents, therefore,not reacting with acidic groups like alcohols.

The first step is the addition of the organocuprate reagent ${\left({\text{CH}}_{\text{2}}{\text{=CH}}_{\text{2}}\right)}_{\text{2}}\text{CuLi}$to the compound (A), which leads to the formation of compound (B), which on further reacting with Grignard reagent $\left({\text{CH}}_{\text{3}}\text{MgBr}\right)$, followed by acidic hydrolysis form the given compound 4,4-dimethylhex-5-en-2-ol.

Therefore, compound (A) is $\text{α,β}$-unsaturated aldehyde that is 3-methylbut-2-enaland adding ${\left({\text{CH}}_{\text{2}}{\text{=CH}}_{\text{2}}\right)}_{\text{2}}\text{CuLi}$(organocupratereagent ) forms compound (B) which is 3,3-dimethylpent-4-enal.

The way compounds A and B react in the reaction is shown as follows:

Compounds A and B