Rule: . . . important that someone DO (bare infinitive verb) . . .
 It is important that the wine (not) be chilled.
 It is important to me, personally, that I improve my English.
Note, According to my teacher, it is important for me to improve my English.
 Last year, I suggested to my son that he improve his English.
Note, the word English is a name, so it should be capitalized.
All the best.