To make things simpler, I've kept existing boot, root and swap partitions and only convert home to LVM, with enough free space for some new volumes.

Of course, this must be done in single user mode, without graphical interface, under the root account:

Before: /dev/sda4 = extended partition, /dev/sda5 = 400Gio home partition

# cd /
# tar czf root/home.tgz home
# umount /home
# fdisk /dev/sda
=> remove partition 5
=> remove partition 4
=> create a new primary partition 4 (the 400Gio available)
=> type 8e (LVM)
# pvcreate /dev/sda4
# vgcreate vgt530 /dev/sda4
# lvcreate -L 50G -n home vgt530
# mkfs.ext4 -L home /dev/vgt530/lvhome
# vi /etc/fstab
=> change the home line
# mount -a
# tar xf root/home.tgz
# systemctl reboot

After: a 50Gio home (really enough) and 350Gio of free space for my future needs (some VM)

For memory, creation of a storage pool for virtualization :

# virsh pool-create-as vgt530 logical --source-dev /dev/sda4 --target /dev/vgt530

From now, it will be possible, during a new VM creation, to use this storage pool, each disk of the VM will be a LV of the host.

# lvscan
ACTIVE            '/dev/vgt530/lvhome' [50,00 GiB] inherit
ACTIVE            '/dev/vgt530/RHEL-5' [10,00 GiB] inherit
ACTIVE            '/dev/vgt530/RHEL-6' [10,00 GiB] inherit

Sorry, but I won't be able to work without  LVM.