Copyright (c) 2001 Chris Huntoon

MOVE - A soldier moves like the chess Rook (i.e., any number of empty cells in a straight orthogonal line).
CAPTURE - There are two ways of capture:
Custodian - If a soldier moves orthogonally next to an enemy soldier and there is another friendly soldier directly beyond that, the enemy is squeezed between the two pieces and captured.
Intervention - If a soldier steps between two orthogonally placed enemy pieces, it captures them both.
The edge of the board may be used to form any line of three to complete a capture. So a soldier may capture an enemy piece by sandwiching it between itself and the edge of the board or by stepping between the enemy piece and the edge of the board. 
It is possible to move so as to capture in more than one direction at a time.
GOAL - A player wins when he stalemates his opponent, or reduces him to two soldiers or less.

According to the author, Cleave is a variant of the Thai game of Mak-yek.

An example

White moves b2 to cell [1] and sieges the black soldiers. If Black moves g7-g5 capturing h5, White moves d6-g6, capturing g5 and reducing Black to two soldiers and winning the game.

There is a ZRF to play Cleave with Zillions.