A Center Controller reported the adjacent center sector told them to descend an aircraft to avoid their airspace without completing the necessary coordination resulting in conflicts with two other enroute aircraft in the receiving controller's airspace..

Narrative

I was working sector XX with higher than usual traffic volume and complexity due to the numerous requests coming from ZZZ1 [ARTCC] for ski country routes and altitudes. I was regularly descending aircraft from FL400 and above to meet these restrictions. I was doing the ZZZ1 [ARTCC] controllers a favor and descending Aircraft X to FL280. It should be noted that this aircraft was landing at ZZZ; an airport for which ZZZ [ARTCC] does not have an LOA (Letter Of Agreement) restriction going into ZZZ1 [ARTCC]. I point the aircraft out to sector XY and initiated a handoff to ZZZ1 [ARTCC] sector XZ. I went about my duties when the ZZZ1 [ARTCC] XZ controller called on the shout line. The controller on [sector] XA noticed I was busy and answered the line for me while looking at my scope. The XA controller sits adjacent to XX. I didn't hear this call; but I was told it was the sector XZ controller telling us to descend Aircraft X below his airspace. The XA controller said unable due to the lack of coordination with the affected from the ZZZ1 [ARTCC] XZ controller.The ZZZ1 [ARTCC] XZ controller called back shortly after which is when I answered.The ZZZ1 [ARTCC] controller requested Aircraft X to descend below his airspace at 2000FPM or greater; saying 'I am unable to get him lower.' I issued the restriction to Aircraft X and coordinated with sector XY. It was then that conflict alert went off with Aircraft Y at FL260; two minutes from the aircraft colliding. Not losing separation; but colliding; as the leader lines were touching and I could not distinguish one from the other. Conflict alert also started going off for Aircraft Z at FL260. I was not watching ANY of these other aircraft as my sector is FL290 and above; and they were outside of my altitude limits and below my airspace. It should be noted that the ZZZ1 [ARTCC] XZ controller would have had both datablocks of the aircraft at FL260 displayed on their scope as limited datablocks; if their altitude limits were set correctly. I eventually coordinated with ZZZ1 [ARTCC] sector XB and stopped Aircraft X descent at FL270. This dangerous and reckless action from the ZZZ1 [ARTCC] XZ controller nearly caused a collision between two or more aircraft. This controller had no place giving a control instruction without accomplishing coordination with all involved sectors or considering all potential traffic; simply because he would have to work hard to get the aircraft a further descent.As the sector XX controller; I should have absolutely said 'Unable' to this request. It was below my airspace and I could not see other involved traffic. I felt pressured to comply because the aircraft was approaching the boundary with ZZZ1 [ARTCC] and I felt like I had no choice but to descend the aircraft or violate ZZZ1 [ARTCC] XZ airspace. The controller at ZZZ1 [ARTCC] XZ needs to read the 7110.65 and understand that they cannot issue a control instruction without accomplishing coordination with ALL affected sectors first. It is dangerous and unsafe and likely illegal to do what they did. It is unreasonable to give a control instruction to have an aircraft miss your airspace simply because there is traffic that can be solved; especially without considering all potential traffic. I sound like a broken record but I'm not sure what else can be done with this controller. On my end; as I said above; I should not have issued this control instruction even if I felt pressured to due to the aircrafts position and past incidents with this area trying to issue control instructions without affecting coordination. I should have left the aircraft stay safely at FL280 and coordinated appropriately with ZZZ1 [ARTCC] XZ.