I live in Lancaster, California, altitude average 3,000 ft.; Summer high temps average 110-110 degrees F for 6 months; Winter low temps average 20DF for as long as 1 or 2 months; average rainfall around 4″, usually in a 2 week period in the late Fall or early Winter.
I’m interested in hearing about a permaculture system in this kind of environment.

How would you put more CO2/carbon into the desert soil?

Deserts are greening from higher CO2, not from more water. CO2 constitues nearly 70% of biomass/carbohydrates.CAM and C4 types of photosynthesis evolved to manage lowers CO2. Lower levels of CO2 in the atmosphere could be among the factors that produced deserts. Plants had to change the way they photosynthesize from C3 to CAM and C4 ( grasses). When you go into mountains you see that plants are becoming shorter- that is because of lower CO2 , not the lack of water.
https://www.csiro.au/en/news/news-releases/2013/deserts-greening-from-rising-co2
Agave and deep rooting mezquite trees are enough to green arid areas.
https://regenerationinternational.org/billion-agave-project
Succulents take water and CO2 at nights when the air is more humid and there is more CO2 close to the ground.
When plants are waiting for CO2 to enter through the stomata they lose water. Higher CO2 saves water.

To covert CO2 into carbohydrates that can be used productively in the soil requires photosynthesizing plants (preferably not in oxidized, aridized, lifeless soil) and water. To “green” a desert, you’re talking about accomplishing this on millions of acres. And, even if you manage to plant and water all those plants, and remediate all that soil, it doesn’t guarantee that you will alter the hydrological cycle of the area enough to support those plants or prevent them from being washed/eroded away. In fact, the only way to substantially alter the hydrology of the desert (which is the only way to truly “green” a desert) is to displace much of it with 1000-year old forests, swamps, marshes, and other wetland ecosystems. In other words, the answer isn’t coming out of a hose.