General aviation Flight Instructor reported a NMAC at a non-towered airport. No evasive action was taken.

Narrative

Events occured at s27; an untowered airport with moderate traffic during peak hours of the day. Training Aircraft X from s27 with student pilot and CFI onboard was flying the left downwind for runway 31. Aircraft Y contacted CTAF freq and announced they would be back taxiing on runway 31. Aircraft X turned left base for 31; and Aircraft Y cleared the runway. Aircraft X turned onto final and completed a touch and go without incident; announcing on CTAF that we would remain in left traffic for runway 31. Aircraft Y departed while Aircraft X was on the upwind from runway 31. Aircraft X turned onto left crosswind approx 1.5 nm from departure end of runway 31. Aircraft Y began nonstandard 45 degree left crosswind around 0.5 nm from departure end of runway 31. Aircraft X turned left crosswind to left downwind while Aircraft Y continued outbound on nonstandard left crosswind; bringing the Aircraft X and Aircraft Y into conflict. Aircraft Y announced that he would not be a factor for the traffic in the downwind; unknown if he had visual contact with Aircraft X. Aircraft X established visual contact with Aircraft Y; who was crossing left to right approximately 300 ft off the nose of the Aircraft X at same altitude. Aircraft X contacted Aircraft Y and advised of the conflict; and requested he fly standard pattern in the future. Aircraft Y departed area to the west.it is the opinion of the writer that the Aircraft Y pilot demonstrated the hazardous attitudes of impulsivity; macho; and anti-authority. first issue was back taxiing on an active runway with traffic in the pattern; potentially creating a conflict with landing traffic when a taxiway was available and empty. second issue was Aircraft Y pilot departing immediately following slower traffic; and seeming in a rush to exit the traffic pattern; deciding to turn an early crosswind and cutting off traffic on the downwind instead of flying the upwind and then departing on the crosswind.