Can you use Nulls as part of a multi-column uniqueness check in SQLite?

Consider a simple self-referencing table in SQLite with the following fields

Create Table Test{
    Id INTEGER PRIMARY KEY, <-- Alias for the RowId
    Name TEXT NOT NULL CHECK(length(Name) > 0),
    ParentId INTEGER REFERENCES Test(Id),
};

CREATE UNIQUE INDEX IX_UniqueNamePerLevel ON Test(ParentId, Name);

We're trying to set a uniqueness constraint on Name for all items which share the same ParentId. In other words, you can have two items with the name 'Joe' provided those items do not have the same ParentId.

The problem is that SQLite seems to treat nulls as distinct, meaning for any level except root items, the constraint works, but you can have fifteen 'Joe' entries all with a ParentId of 'null.'

Bonus points if you can show how to make that constraint trim leading and trailing whitespace on insert/update, and ignore case for the uniqueness constraint too.