So, as I discovered the hard way, equipping an elemental weapon actually reduces the effectiveness of your elemental skills.

Using an elemental skill does not replace the element of the attack, rather, it makes a hybrid attack. So using Electric Blitz while equipped with a Chirijiraden will result in an attack that is 50% electric, 50% fire. (Source: https://exviuswiki.com/Mechanics#Elemental_Resistance )

This means even if two Lightnings use the same Electric Blitz and one has an elemental weapon, THEY MAY FAIL TO CHAIN if one Lightning triggers a Weakness popup and the other does not (must have at least 50% modifier to trigger weakness popup, but using a 50% electric attack against a -50% electric resist enemy results in only a 25% modifier).

I first noticed this when I was successfully creating elemental chains against some enemies but not others, here's the video:

https://www.youtube.com/watch?v=4vk5RvoVpFo&feature=em-upload_owner

For equipment, my Lightning has Chirijiraden, while the other Lightning has Excalibur and Machine Killer.

At 0:30, the Electric Blitz attacks successfully chain against the top two enemies, as no weakness are applied and both attacks have an electric element. However, against the Pantheron, the elemental chain fails, as the friend Lightning triggers the warning popup (and damage delay) with her 75% modifier ( [50% electric weakness 0% light weakness] / 2 50% machine killer = 75%) while my lightning does not with her 25% modifier ( [50% electric weakness 0% fire weakness] / 2 = 25%).

At 1:00, the same thing happens again against the Beta Behemoth (same math as above).

While it seems silly that two attacks hitting at the same time will not chain if one hits a weakness and the other does not, but that is how it is.

TL;DR Elemental weapons weaken Lightning's elemental attack skills. If you can, stick to non-elemental weapons unless the enemy you are fighting has a specific weakness. For this event, you can still trigger the weakness popup by using Machine Killer to avoid your damage getting out of sync.

Edit: Multiple users have pointed out that another strategy is to ensure both lightnings have the same elemental and use Area Blast/Crushing Blow. Just make sure if you do this that either A) Both Lightnings will trigger the weakness popup (single element or both have machine killer), or that neither do (multi-element without Machine Killer).