In American English the following sentence would be correctly written:

It is true: She wanted me to say she was with me when her husband was killed.

The first word after the colon is written capitalized because it is part of a sentence.

Also this sentence is correct, since what follow the colon is not a sentence:

I did all I had to do: buy the train tickets, pay my taxes online, and order dinner from a take away.

How would the equivalent Italian sentences be written? Would the word after the colon be capitalized in the first sentence, but not in the second one, or would the first word after the colon be written all in lowercase?