Can we assign a self IP for the external VLAN in the same management VLAN.

standard - the management interface/configuration is managed differently internally (linux manages management IP, management routes, etc.) than self-IP addresses (F5 TMM manages switch-ports and self-IPs).

the closest "official F5" note i can find regarding this is https://support.f5.com/kb/en-us/solutions/public/13000/300/sol13342.html "The IP address assigned to the management interface must be on a different network than the self IP addresses assigned to VLANs."

It is possible to manage the device via self-IP addresses, but you will want to take care in locking down those self-IP addresses as much as possible, and you still must have unique management IP addresses configured among devices in an HA pair as those addresses are used as device-identifiers by the F5s.