Does the investment component of GDP (demand side) identity include expenditures by the government and households? In the identity GDP=C+I+G+(X-M) does "I" include expenditures by the government and households?

For example, if a construction firm builds new residential properties and a household buys this property. Is the value of the new residential property allocated as 'firm' expenditure or 'household' expenditure and where in the identity above would these expenditures fall?

Also if a government builds a school or bridge would this go under "I" or "G"?
“I” would be called private domestic investment. (https://apps.bea.gov/iTable/iTable.cfm?reqid=19&amp;step=2#reqid=19&amp;step=2&amp;isuri=1&amp;1921=survey)   

Firm and household expenditures for investment both count as private investment. 

(https://courses.lumenlearning.com/boundless-economics/chapter/measuring-output-using-gdp/)

Any investment expenditure by a government would be categorized under government spending. 

If a government builds a school or bridge it would go under government spending “G”. (https://courses.lumenlearning.com/wm-macroeconomics/chapter/calculating-gdp/)