Do airline SOP normally recommend an in-flight engine restart in the case of an unexplained shutdown?
Your question mentions the 737 or similar. If you want answers limited to 2-engine jets, let me know and I'll remove this answer as it is for the 747-100 or -200.

Whether to restart or not depends on the nature of the failure. If there appears to be no problem to do otherwise, a restart would be in order. The checklist for an inflight start from the last airline I flew for (now out of business) is below. The code to the right of each action is who performs it: C - captain, FE - flight engineer, All - all 3 crew members.

Thurst Lever ............................... CLOSE        C

Start Lever ................................ CUTOFF       C

Fuel Pressure .............................. AVAILABLE    FE

Bleed Air Valve ............................ OPEN         FE

Engine Ignition (Sys 1 and Sys 2)

     Above 250 KTS ......................... FLT START    FE

     250 KTS and Below ..................... GND START    FE

Start Lever ................................ RICH/IDLE    C

Engine Instruments ......................... STABILIZED   ALL

Start Lever ................................ IDLE         C

Engine Ignition ............................ AS REQUIRED  FE

Electrical Power ........................... RESTORE      FE


The most likely cause for a Pratt &amp; Whitney JT-9 engine to quit is poor technique on the part of the pilot. If you're at altitude and starting the initial descent, you should put the nose down before bringing the thrust levers back. Also, don't slam them back, take a few seconds to do it, especially if you're above FL370.

It's worth noting that on the 747, the loss of an engine (no restart) enroute is not considered a serious emergency. At the two 747 carriers I flew for, it was the captain's discretion as to whether to divert or to continue on.

Also, it's not always the case of either the engine works at the power of the other engines or it doesn't. There are situations where the engine might be available, but with partial power. As I remember there were at least two such cases, one of which was high oil temperature. Basically the work-around was to run the engine at that power setting that would keep the oil temp below the maximum.