Danger to Expose Eye to 10,000+ Lux of Ultraviolet Light Is it dangerous to expose the human eye to 3,000 - 10,000+ lux of UV (ultra violet) light in the 380 - 400 nm wavelength? 
I understand anything below 400 nm is tech classified as UV/UVA radiation but it is very close to the violet spectrum (which begins at 400nm). Further, most people can see light at 380+ nm ("the visible range or light spans 380 to 780 nm") which means that if it was damaging to the eye it would likely activate my eye's natural reflex? 
Further, my belief that this would not pose a danger is partially based on  this Wikipedia article  which states that "UV radiation constitutes about 10% of the total light output of the Sun, and is thus present in sunlight" It also states that "at ground level sunlight is 44% visible light, 3% ultraviolet" and that "more than 95% is the longer wavelengths of UVA". On a bright sunny day more than 100,000 lux of light reaches the earth from the sun". A quick calculation shows that there's at least 3,000 lux of UV light at ground level which includes UV light from 315 to 400 nm. 
Hence, my logic is that if we're talking about UV light with an upper wavelength of 380 - 400 nm, 3,000 lux should certainly be acceptable but probably more is also fine. But what is the upper limit?
Basically, I'm looking for an authoritative answer to this question (hopefully with sources and explanation): what is the max amount of UV light I could safely expose my eyes to in the 380 - 400 nm wavelength? And for how long?
Clarification: Based on @The Photon's response I think the correct word I'm looking for is "irradiance" not lux.
My question is: how much light in the violet spectrum (380 nm - 400 nm) is it safe to expose the eye to. In this case I would define light in terms of the amount of watt used to produce the light (for a lack of a better term). 
For example, if I sit within 3 feet of 30 50W Ultra Violet UV LED would it be dangerous? Why / why not?
 A: 
according to Wikipedia the sun emits about 10% light as UV (the spectrum of sunlight on earth during a typical day includes a continuous distribution of wavelengths from approximately 300 nm to approximately 1200 nm) and on a bright sunny day about 100,000 lux reaches the earth from the sun. This means that about 10,000 lux is in the UV spectrum

This is incorrect reasoning.
First, even if the sun emits 10% of its energy in UV, much of this UV is filtered out in the ozone layer and atmosphere, and does not reach the ground. So the fraction of UV in solar radiation at the ground is much lower.
Second, the unit lux measures luminous emittance, meaning how bright a source appears to the human eye.
The 10% of the sun's radiant flux that falls in the UV does not contribute (much) to its luminous flux because the human eye does not see it. 
In order for a UV source to produce 10,000 lux of illumination, it would have to be so high-powered that it looks as bright as ordinary daylight. This would be very powerful indeed, because our eyes do not detect UV radiation very strongly at all. 
Very likely this source would produce burns immediately if human skin were exposed to it. Damage to the eyes would be even worse.

at ground level sunlight is 44% visible light, 3% ultraviolet ... A quick calculation shows that there's at least 3,000 lux of UV light at ground level which includes UV light from 315 to 400 nm.

Again, this does not follow. The statement that sunlight is 3% ultraviolet is referring to radiant energy, not illuminance. That is, it is saying 3% of the energy received from the sun is in the UV. That does not mean that UV is responsible for 3% of the illumination perceived by our eyes. 
A: Since the human visual system has this habit of focusing on things, technically the unit you are interested in is radiance (https://en.m.wikipedia.org/wiki/Radiance), which is measured in units of power per unit of solid angle per unit area. This is part of why lasers are so dangerous. But if you can assume that the source of light is diffuse/lambertian, then knowing just the irradiance at the surface where the light is emitted will suffice (if the outgoing light is evenly distributed across a hemisphere of directions, this is 2 steradians, allowing you to compute the radiance).
In this case you have 50 watts of power being emitted from a small array of leds at the center (maybe 1 inch square?) and then a reflector to help redirect some of the side light forwards. Given that it's on the consumer market it's probably generally safe, but I still wouldn't stare directly at it for a long time. The pupil's natural reaction to constrict in bright light can be much less with violet/UV light since since your cones are much less sensitive to it.
But let's say you were the one manufacturing the light and you wanted to know if it met certain regulations for eye safety. You might consult the
ICNIRP guidelines on limits of exposure to incoherent visible and infrared radiation (https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.icnirp.org/cms/upload/publications/ICNIRPVisible_Infrared2013.pdf&ved=2ahUKEwitnO3j28ntAhUH1VkKHdRMDogQFjAQegQICRAB&usg=AOvVaw1mSSlMGJk_ynGLMa7JZ302). You'll notice that the radiance limit depends on how long the exposure lasts (which makes sense -- even an extremely bright flash, if it lasts only a few nanoseconds, can still be considered safe).
So back to our example: Say we have a 2.5 cm square surface (0.000625 square meters) putting out 50 watts over a solid angle of directions of 2 ≈ 6.28 steradians. The radiance is about 12,700 watts per square meter per steradian. At a distance of 1 meter, the apparent size of the light to the eye would be approximately the same number of radians as size in meters: about 0.025 radians. This would be considered a medium size source according to the paper. For exposures longer than a quarter second, the limit would be 28,000/ 0.025 = 1.12 million W m^-2 sr^-1. For other quantities like irradiance, usually we'd have to do a bunch more math to figure out that quantity at the eye, but radiance is special. There is a law in optics called the "law of conservation of radiance" that tells us that it will be the same (assuming no absorbtion by the air or the lens of the eye, which is approximately true here). So at 12.7 thousand W m^-2 sr^-1, we're well below the limit. Similarly, the limit on the total dose works out to be 284,000 J m^-2 sr^-1, so theoretically you could stare at it for about 22 seconds and still be below the limit. That's of course given that all the assumptions I made are correct.
Still, if it were me, I wouldn't try it.
