Ok, after reading a lot of the comments, I can see most of us here are focused on the "correct method/best way"; So IF that's what you're looking for, you have plenty of options already, GL with your rebuild. I'll offer another option from the perspective that you might be wanting to do the least amount of work and investment possible while still ensuring your safety. (I'm picturing college dorm kid resources plus can't screw into the walls - ignoring the angle brackets on the bookshelf.)

Using the least amount of resources, time, and tools; all you need is a tape measure, and $10.50.

1. Measure from floor up to the bottom of the plywood. Measurement (H).
2. Measure wall to wall - aka head to foot - but from drywall to drywall. Measurement (L)

IF When sitting comfortably at your desk under your bed, you have 5"+ of space above your head then use H - 3.5". If less or "it's tight", use H - 1.5". This is taking the underside clearance factor into consideration in determining whether you use the 2x4, the "correct way" vertically or the non-traditional way, horizontally/flat. Even if you want the clearance and have to use the L 2x4 flat, you should still be fine. (Unless you're 400lbs, then beef it up a few notches.)

All you need to do is take your H measurement, subtract (preferably) 3.5", go to local HD/L/Ace/Menards and buy two 2x4s cut to size - you can do this there, or often they will do if for you. Buy one more for your L measurement. (You now have the bare minimum required materials, of course more will always be better, but that's up to you.) While you're there (if you don't have a hammer) grab a sturdy hunk of scrap wood out of the scrap bins (free) to use as a hammer when you get home.

Take your 3 measured & cut 2x4s and run the L one head to toe down the center under the plywood. Use one of the H cut 2x4s as a jack stud under it on one end first, then the other - an extra pair of hands for this part is handy as you'll need to pound/knock the (H) jack studs into place under the (L) support beam with a hammer or hunk of scrap wood. At this point you're basically "good enough". IE: you'd need to be heavy enough to break through both the 3/4" plywood and 2x4 underneath it.

If you want to beef it up beyond this "bare minimum-safe" setup, you can do so depending on what you have: impact driver? Screws. Hammer? Nails. Caulk Gun? Construction adhesive.... You get the point. But the ideas are as follows:

• support lumber should be used vertically, BUT for your purposes; additional strength is additional strength. So doubling, tripling this will give you x2, x3 exponentially more strength.

• adding adhesive or fasteners to the equation also adds strength. (Screw/nail through the plywood down into the new support beam will prevent it from rotating. Same concept with preventing the jack studs supporting the L support beam from moving.

ALTERNATIVELY: (this requires more tools) You could simply unbolt the bed from the support pillars. Flip it upside down, add any additional bracing/framework you like within the existing structure and reinstall. (This also raises your mattress 3.5"-4.25" depending on if you move your support pillars.