Lucena Position - No cut-off?

Hello,

the Lucena Position is well-known, however apparently not by myself. I stumble across the following defense by black:

Black's rook has occupied the line of its own king, making it impossible to White to cut-off the enemy's king more than just one line (without the white Rook on the d-file: Rd2+). To my understanding the Lucena Position is nevertheless based on the idea that first and foremost I have to get the opposite King trapped as far as possible from my K+P. How to commence from this position as White? Kg7 does not seem to benefit White as Rg1+ chases my king back in front of my pawn. Alternatively, just 'building the bridge' not minding the proximity of the enemy-king does not work either - the eventual exchange of rooks on f4 gives black the chance to grab the final pawn and finalize the draw.

What's the trick I miss? It is certainly obvious and simple, however, I just don't get the solution so far.

Cheers

No need to Lucena here: Ke8 wins ...

Hi,

and thanks for your valuable input!

@Andreas: This indeed was simple and swift - and does the trick! Thanks for pointing that out for me, I kept missing that move. 

@Viktor: Thank you for the insightful elaboration! Things become slowly clearer to me. Correct me if I'm wrong, but it seems that as long as the black rook is not covering the adjacent g-line (as in your 8... Rg3 alternative), no Lucena is needed. Lucena seems to come in place only when both K and R take away all room for manoeuvre from the white King.

Then again, if your suggested position is changed so that the black rook still covers the king's e-line but stands now additionally in front of/closer to the action than the white king (means e.g. black King d2 while white King e1), is it enough to bring the white rook just closer to the action than the black rook is? The aim would be to transpose into the position as illustrated by Viktor before. The changed position of the rooks would, if I get the situation properly, lead to following play:

Is this correct? Do I miss something again?

Thank you a lot

Nope. From there just 1. Rg1 Rf2 2. Kg7 Ke7 3. Re1+. Same idea as before.

No need to try and make everything transpose to the Lucena. If there's a shorter win, take it. The Lucena is the result only when black does the best defence.

Thank you Remellion, this sheds light on this issue! Again it looks now neat, nice and obvious, but it kept being unattainable to me. The best defence for black would indeed be Rg2 (and the Lucena finally arises), right?