I think there are three main conditions 1) Not affected by long periods of time 2) That it does not transform phase with a change or an increase in temperature 3) That it does not change phase in the process of oxidation and reduction
In general, to check the electrochemical stability of electrode/electrolyte, one can opt for the three electrode system study where you have to use one working electrode, reference electrode and counter electrode.
As in your case, you want to check the electrolyte stability on Lithium metal, so lithium metal would be your working electrode, you can use Ag/AgCl as the reference electrode and Pt wire as the counter electrode.
In my understanding, by doing the suggested experiment, you will get the information regarding stability of your working electrode and electrolyte.
1. High rate and stable cycling of lithium metal anode https://www.scienceopen.com/document_file/d3a728b8-e06e-42a8-9354-e67d7fa3d081/PubMedCentral/d3a728b8-e06e-42a8-9354-e67d7fa3d081.pdf
2. Toward safe lithium metal anode in rechargeable batteries: a review https://pubs.acs.org/doi/abs/10.1021/acs.chemrev.7b00115
3. Design principles for electrolytes and interfaces for stable lithium-metal batteries https://www.researchgate.net/profile/Mukul_Tikekar/publication/307949534_Design_principles_for_electrolytes_and_interfaces_for_stable_lithium-metal_batteries/links/5a2efb33aca2726d0bd6a80a/Design-principles-for-electrolytes-and-interfaces-for-stable-lithium-metal-batteries.pdf
4. Reviving the lithium metal anode for high-energy batteries http://web.stanford.edu/group/cui_group/papers/Dingchang_Cui_NATNANO_Review_2017.pdf