According to Finn's Thermal Physics Third Edition, the First Law of Thermodynamics is described as:
If a thermally isolated system is brought from one equilibrium state to another, the work necessary to achieve this change is independent of the process used.
Which I think must be a rephrasing of the law of energy conservation. However, work is an inexact differential, in that the work done in a system transitioning reversibly from $(P_1,V_1)$ to $(P_2,V_2)$ is entirely dependent on the path taken. But here the quote sounds like its contradicting the previous sentence. The textbook goes on to say
The statement above is equivalent to saying that the adiabatic work $W_{\text{adiabatic}}$ expended in a process is path independent, ...
which makes some sense to me, I think. If the walls of the system are insulating, then no work on the gas can go to heat. However, how does this fit in with the general concept that work is path dependent? Are they saying there is always a minimum amount of work necessary? Because a bunch of paths can be made from $(P_1,V_1)$ to $(P_2,V_2)$ which don't necessarily expend the same amount of work.