I ́d like to write an eMail with my second account. But I cannot choose between my accounts. Any idea?

Hot to choose between accounts to write eMail?

When you create a new email  click on the senders name.  If you have setup other address they will appear in the drop-down

…Or, to create a new email with a specific email account without selecting addressees, simply select the _ Inbox of the email account _ you want to use and then click New in the task-bar above it but below the  File menu.  When a new email window appears it will have your email address for that account already in the From area.

So, to simplify, select the email account first, the inbox, then New to compose your email.  Then add addressees in the To line as you wish.