JERUSALEM — Hamas, the militant group that runs the Gaza Strip, has been trying for over a year to keep a lid on its conflict with Israel, to improve the abysmal quality of life for the two million Palestinians under its control, and to keep millions of dollars in cash coming in each month from its generous allies in Qatar .

But a nettlesome, unruly and heavily armed little group called Palestinian Islamic Jihad has repeatedly sabotaged those plans by firing rockets at Israel, which more often than not has responded by raining down destruction on Hamas’s own installations and men.

On Tuesday, impatient with Hamas’s failure to curtail the group, Israel assassinated a top Islamic Jihad commander, a maverick said to be responsible for nearly every instance in the past year in which an incipient cease-fire on the Gaza-Israeli border was wrecked by violence from the Palestinian side.

The killing put Hamas in the uncomfortable position of sitting on the sidelines while its much smaller but more militant rival exchanged blows with their shared hated enemy over two long days of battle.