English Grammar

[Gilles]

Dear Teacher.
English is not my mother tongue since I am a francophone. I am experiencing difficulties with respect to the subjonctive. I would appreciate your reviewing and correcting the following sentences for me.
First example:
"It is important that the wine is not chilled" or would it be preferable to say:
"It is important that the wine should not be chilled " ?
Second example:
present tense: It is important for me that I improve my english.
pas tense: Last year I suggested to my son that he impove(D)(*) his english
(*) is the "improved" written correctly ?
Thank you very much for your time.
My e-mail address is
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[Casiopea]

Welcome, Gilles.

Rule: . . . important that someone DO (bare infinitive verb) . . .

[1] It is important that the wine (not) be chilled.

[2] 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.

[3] 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.