It looks like you are trying to use "lilo" on the second hard disk "sdb". The BIOS usually doesn't boot directly to the second hard disk. How are you planning to boot the second hard disk? Are you going to change the BIOS default boot disk, or are you going to chain to the second hard disk from a boot loader already installed on the first hard disk?
When you change the BIOS default boot disk you may find that the drive assignments change. To boot by changing the BIOS default disk, you should install LILO with the default boot disk in the BIOS set to the second hard disk. That will probably make it appear as "sda" and not "sdb". Always install boot loaders with the BIOS configured as it will be when you are going to use those boot loaders. If you're not going to change the BIOS default boot disk then none of this paragraph applies to your situation.
If you're chain booting to the second hard disk, then make sure that you install LILO to the MBR of the second hard disk "sdb" or the partition boot sector of "sdb2". Then configure your existing boot loader to chain to that boot sector. Although you can install LILO to the MBR of "sda" that will require that both "sda" and "sdb" are present in order to boot anything.
Do you have an existing LILO or GRUB boot loader on "sda"? If so, you can load Slackware using that boot loader and avoid chaining to LILO on the second hard disk.
It will be helpful if you can post the contents of your "/etc/lilo.conf" file. To do that you will need to boot some operating system. You may find it easier to use a Linux boot CD with a graphical environment, such as Puppy Linux. Use the boot CD to get on the Internet and post the "lilo.conf" file.
You can also type out or edit "lilo.conf" using the Slackware installation CD. Boot the Slackware CD. Then type in commands to mount your partitions and change the root to the OS you are installing.
mount /dev/sdb4 /mnt/hd
mount /dev/sdb2 /mnt/hd/boot
mount /dev/sdb3 /mnt/hd/var
mount -t proc none /proc
mount -t sysfs none /sys
To type out "lilo.conf" you can do this.
more < lilo.conf
To edit "lilo.conf" do this.
To re-install "lilo" with a modified configuration type in the "lilo" command.
When you have finished making changes un-mount the partitions and restart the computer.
You mentioned a thumb drive. When you boot from a thumb drive, the BIOS may make that "sda". That could cause a problem because when you remove the thumb drive, the BIOS will change the drive assignment of "sdb" to "sda". I'm not sure if LILO will work correctly. You may have to use GRUB, or patch the LILO boot sector with the correct BIOS drive ID (80 hex instead of 81 hex).