The green and yellow wire is a ground. You can ignore the ground wire and just use the blue and brown wire as you would do for the other lightbulbs or run a separate ground wire to the light.
The Arduino board does not have an earth ground. The Arduino has a common being used as a ground. I'd run a separate earth ground wire to the light. I'd not use the common ground of the Arduino as a earth ground.
Complete misunderstanding of the term "ground" in the Arduino context. It has absolutely nothing whatsoever to do with the actual ground wire in a mains installation.
It may happen to indirectly connect via the power supply, but the "ground" in an Arduino circuit is a simple arbitrary shorthand for "common negative return".
Brown should go to your relay. A black wire should go from relay to light bulb (may also be brown). The blue lead can be connected to the other end of the bulb. The blue one is supposed to be at 0 V!
At least that is the standard in NL and I think also EU. In USA colors are different.
Don't forget to ground/earth the light (it need a ground because it have a metal case). And while you are at it, you can earth/ground your arduino, too.
Well, that's not the primary reason.
Unless you put the arduino in a plastic box (which you should), and every part of it is non-metal (including possible switches, connectors, future connected equipment, it is regulated to require grounding. And, frankly, all phone-chargers and a lot of power brick violate this.
It is regulated that anything you can touch has galvanic separation from mains (a proper transformer or relay or optocoupler plus ssd can provide that) . Including sufficient insulation/distance. There is no regulations for grounding.
The charger by and of itself have plastic exterior so under absolutely no circumstance can the user touch the case and get shocked. Yes.
However, when you plug your phone in, your phone is considered as "one" with the charger. And because your phone has a metal case, it violates the double insulation requirement.
I have a Samsung metal-bezel display with a external power adapter (with grounding) of 12V 3A attached (cannot be removed). The charger is "double-insulated" (plastic cover and more than enough clearance), but there is no "double insulation" anywhere. In fact there is only two: "do not disassemble" and "electric hazard". (among the other boring regulatory ones like UKCA and CE)
Because the metal-bezel display has a metal cover and railing. And when they are attached, it is not double insulated.
Will you be fine if it is not grounded? absolutely. No matter if it is considered as "double insulated" or not.
But if you can, why not?
Not sure you comprehend what "double insulated" means.
If (and that's a an "if") the power adaptor is double-insulated, it means there are two layers of insulation in every part between the mains and the secondary. So in the transformer (and they all have one) there are two insulating layers between primary and secondary.
The insulation on the wire itself does not count. Except in China - we know a lot of the crap sold is anything but double insulated.
But if it genuinely is certified - we assume what is sold by reputable dealers is - then there is no problem with exposed parts on the secondary side and no requirement whatsoever for grounding. Just as well as the connectors on laptops and to a lesser extent on phones, are effectively exposed.
Because it causes severe ground loop problems in audio equipment and actually, a safety hazard in the remote situation that there were to be some exposed "live" mains nearby.
(Just on my way to church now where I do have exactly such ground loop problems! )