Greetings Poldom65,
Apple Support Communities is a great place for help with using your Magic Keyboard and Split View on your iPad Pro. You’ve walked through great troubleshooting steps thus far by restarting your iPad, and confirming you’re running iPadOS 15. There have been several updates to iPadOS 15 since the initial release. Have you verified you're running the latest iPadOS 15 version? If not, you can find the steps to check for and perform available updates in the following article.
Update your iPhone, iPad, or iPod touch
In addition to confirming the iPadOS version, have you verified that you’ve tapped in the field you would like to type in within the desired application when using Split View? If not, try that step next to see if this helps isolate the issue.
There are some features and functions that may require use of the onscreen keyboard, and may not be supported via the Magic Keyboard. This information can be found in the Switch to the onscreen keyboard section of the following article.
Set up and use Magic Keyboard for iPad
Let us know how that goes!