@floribert - Re-installing Lion is a bit tricky depending on the system you have.
Assuming its a newer model that allow you to boot up under USB and your drive is setup as a bootable drive with Lion you should be able to select it using the Startup Manager holding the Option key
Reference: Mac startup key combinations
That will allow you to select your external drive to boot up from. But that still won’t fix your internal drive. This gets into overcoming the certificate expiration issue. Here we can’t get a fixed installer as Lion is no longer being supported. So we need to trick the installer telling it its back in time when the OS was still active by dating the systems clock back in time - 2012!
That will allow you to run the OS installer!
But you still may have issues with the drive and likely need a new drive. Here we do need to be careful as most of the drives sold today are fixed SATA III (6.0 Gb/s) so depending on your system you might require a SATA I (1.5 Gb/s) or SATA II (3.0 Gb/s) drive as its systems SATA port can’t support the faster SATA III drive.
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Please tell us what the system is (model, size & year) as I transferred your question onto its now question.
crwdns2934271:0crwdnd2934271:0 Dan crwdne2934271:0