Why isn't ElGamal a symmetric-key scheme?

I was asked about this question by a friend. He considers that Alice and Bob share a secret key s=g^(xy). So why isn't it considered a symmentric-key scheme? According to Wikipedia:

Symmetric-key algorithms[1] are algorithms for cryptography that use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext. The keys may be identical or there may be a simple transformation to go between the two keys.

I observe that the shared s is quite easy to compute by Alice and Bob.

I know there's misunderstanding, but I can't convince myself now.

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• Going from g^(xy) to m is only part of ElGamal. ​ ​ – user991 Nov 16 '16 at 9:28
• Yes, but they do share the same g^(xy) at some point of encryption and decryption, right? I know the shared material doesn't make ElGamal symmentric, but why? – Donald Wu Nov 16 '16 at 9:39
• Yes. ​ The shared material isn't ElGamal's decryption key. ​ ​ ​ ​ – user991 Nov 16 '16 at 9:40

g^x is the public key of Alice and x her private key, while g^y is the public key of Bob and y his private key.
s=g^(xy) is the shared secret between Alice and bob, that can only be computed by them.