Halliday & Matthiessen (2014: 516):
The projected idea clause is either an indirect statement or an indirect question; … different sets of verbs are associated with these two types. In the environment of ‘mental’ projection, the contrast between statement and question is not concerned with the speech functional orientation of giving vs. demanding information but rather with the status of the validity of the information. In a statement, it is fixed with respect to the polarity and the elements of transitivity (realised by an indirect declarative clause optionally introduced by that), but in a question, it is open with respect to the polarity (realised by an indirect yes/no interrogative clause introduced by whether or if) or one (or more) of the elements of transitivity (realised by an indirect wh- interrogative clause introduced by who, which, when, where, etc.). Consequently, mental clauses representing an ‘undecided’ state of mind are used to project indirect questions. These include clauses of wondering and doubting, finding out and checking, and contemplating, which tend to be characterised by special lexical verbs such as wonder, ascertain; for example:
||| Israeli Prime Minister Ehud Barak said Tuesday || he doubted || President Clinton could broker an Israeli-Palestinian peace deal before the end his presidential term on Jan. 20. |||
||| I’ll ask Jenny about laptops || and find out || whether we have got any. |||
||| She had not been in || when he had phoned || to check || whether they were going out for dinner that night. |||
||| It should be noted || that the first step taken by the underwriter or agent is [[[ to examine the policy || to ascertain || whether the loss is recoverable thereunder]]] . |||
||| He investigated || whether his feeling [[ that the veena produced the most exquisite musical sound]] was a subjective reaction || or has a sound physical basis. |||
||| Let us now consider || whether the arrangement of the stanzas in the particular order bears out any such meaning [[ as we have got from it]] . |||