"I know that you are rich."
Translation:So che sei ricca.
"Conoscere" is used for persons, places, things, whiles "sapere" is used for facts.
When in doubt (but this is a rule I am making up right now, so there might be exceptions I haven't thought of!) if you can say "I know THAT", then a safe bet is to use "sapere che" (conoscere isn't generally followed by che).
Good question - why is this not the subjunctive?
Probably because sapere implies certainty, whereas for example pensare does not and would want sia. If someone next tells us that sapere does not imply certainty, then the reason could be that this sentence appears before Duo's tree reaches the subjunctive. :-)
Because the "Lo" in this context refers to "it", it gives the sentence a subject. For example you say "Lo so" to mean "I know (it)" However in this example we already know what "it" is; "it" = that she is rich.
Your example would translate in English as "I know IT that she is rich", the "it" is unnecessary in the English just as the "lo" is unnecessary in the Italian version. I hope that helps explain it! In bocca al lupo:)
Because 'quello' isn't the right 'that' for this occasion. There is the that that indicates which thing you are talking about and there is the that that is used to join bits of sentences together. Quello is the first of those. It is either a demonstrative adjective or a demonstrative pronoun. So you either use it with a noun that would use 'lo' as its definite article, or you use it with an adjective.
But if you want to say "I know that you are rich' you need to use 'che' for the 'that' since the 'that' there is being used to join the bits together rather than to demonstrate which object is being talked about. Conjunction. That's what it is, 'che' is used for the 'that' that is a conjunction.
See the rest of this thread. What may be "good style" in English is not necessarily good style in any other language. This is how you say it in Italian---with the "che." It is necessary. It USED to be necessary in English, to distinguish reported speech from direct speech, but now we have lost that distinction, so you cannot tell in English whether a quote is direct or a paraphrase unless actual quotation marks are used.