the omission of an implied "の" creates the appearance of a 四字熟語{よじじゅくご}?

Question

I believe that www.yojijukugo.com has a pretty thorough list of 四字熟語? Is there a better site?

However, I know that I've seen a lot of 4, or more, adjacent kanji that are not listed as 熟語。An example is "原子発電{げんしはつでん}"。

1. When I see 4 adjacent kanji that are not listed as 熟語、then that just means "の" is being omitted (to save space when printed). From context, you know the meaning without the "の".
2. You say "原子発電" instead of "原子の発電" just as an extension of what is done when writing.
3. real 四字熟語 are pretty rare, so you must assume that 4 kanji in a row is not a 四字熟語 until you can prove it in a dictionary, right?
4. The ability to omit "の" when, by context, everyone knows it is there, allows you to say stuff like "愛国精神{あいこくせいしん}", "愛社精神{あいしゃせいしん}", "愛人類精神{あいじんるいせいしん}", etc. right? The technique is, if the "の" is obvious by context, you don't necessarily have to say it?

Answer 1

As for your last question, please read this question and answer first:

Can kanji compounds be formed arbitrarily?

This question is about two-kanji compounds (熟語 in the narrow sense), but the answer there is basically true for many cases involving four kanji or more. In fact, most of the "四字熟語" you encounter, like 火力発電 or 愛国精神, are established set phrases, and larger online dictionaries like Eijiro list many such phrases. Don't try to freely combine arbitrary two-kanji 熟語 (in the narrow sense) before you definitely hear that combination. We have 学校生活 and 学校新聞 but we don't have 学校屋根 or 学校授業.

But there are exceptions. You can combine multiple 熟語 (in the narrow sense) without particles in the following cases. (By the way, some people may say such long compounds are 複合語 but not 熟語)

1. When space-saving is really important, as in news headlines. Today's Yahoo! headline has "中国高官 続く自殺に粛清の声" and "FW武藤 欧州強豪が極秘視察", but they are usually "中国の高官" or "[FW]{フォワード}の武藤を欧州の強豪が極秘に視察" inside body texts and daily conversations.
2. When creating a proper noun, something like an organization name. 世界自然保護基金日本委員会, 全日本高校生英語弁論大会関東予選, 北海道大学医学部附属病院根室分院内科研究室 etc.
3. When you want to introduce/define a new concept, and use it many times. You can use 石油資源不足国 as something like a set phrase when you are writing an extensive article about oil-poor countries. But if you just want to say "Japan is an oil-poor country" once, then "日本は石油資源が不足している国だ" is better. You can't generally say 母親電話 to refer to "a phone call from my mother", but if your company created a new service then you might name it as 母親電話 (Mamaphone?).

And you may have to learn which 熟語 tends to be a part of longer 複合語. "～精神 (～ spirit)", "～指数 (～ index)", "～作戦 (～ mission)", "～装置 (～ device)" are the examples of such "productive" 熟語.

Answer 2

What makes something 四字熟語 is not very easy to define. What you are talking really comes down to a concept known as idiomaticity in Linguistics.

Having that said, 四字熟語 are generally words used as idioms (i.e. 慣用句). A 慣用句 is an expression that often utilizes more than one unrelated words to mean something completely different. For example, 画竜点睛 means something crucial that is necessary to make something complete, but it is not straightforward to derive this meaning from 画竜 and 点睛 separately.

On the other hand, words like 原子発電 are constructed words (or 複合語). They differ from 慣用句 that they are artificially put together to mean a concept derived from those words, in this case, 発電 using 原子. Here are some other examples that are not considered as 四字熟語: 株式会社, 高等学校, 関東平野.

As a side note, a lot of 四字熟語 used in Japanese originally came from Chinese, and there is usually an interesting story behind how each of them came to use. Some of my favorite ones are 羊頭狗肉 and 矛盾.