Self-checking pieces

A self-checking piece is a piece that can give check to it's own King.
To put otherwise: a move is illegal, if it leaves your King in the attacking range of your own piece with this self-check ability.

Two points just to avoid confusion:
The self-checking piece can still also capture the opponent's pieces, or give check to the opponent's King.
The self-checking piece cannot actually capture friendly pieces.

Example 1:

A self-checking Knight, here visible with a red dot.
1. Nc7# because after e8xc7 the black King is still in check.

Example 2:

A Knight with self-checking King's move, here visible with a red dot.
1. Qxh7# because after f6xh7 the black king is still in check.
(So possibly, we define a piece with a self-checking move, that does not coincide with the normal move.)

The point of introducing these pieces might be, that it can potentially introduce a new area of strategy and tactics.
Because one must maneuver around one's own self-checking threats, and exploit the same weakness in the opponent's army.